#513035

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂²×D₁₆
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,26)(12,19)(13,21)(14,40)(15,45)(16,47)(17,49)(18,25)(20,31)(22,42)(23,44)(24,41)(27,59)(28,55)(29,65)(30,54)(32,50)(33,61)(34,60)(35,67)(36,53)(37,68)(38,69)(39,58)(43,57)(46,56)(48,70)(51,63)(52,62)(64,72)(66,71);(2,5)(6,8)(9,11,15,28,16,29,43,35,13,27,38,62,39,63,14,12,18,31,42,34,48,66,37,32,17,30,46,64,41,33,44,36)(10,19,23,51,24,52,56,59,49,67,68,65,70,55,22,26,25,53,40,61,58,72,69,54,21,50,57,71,47,60,45,20);(1,3,4,7)(2,6,5,8)(9,12,14,33,41,62,38,30,17,35,43,66,48,28,15,31,18,36,44,63,39,64,46,27,13,32,37,29,16,34,42,11)(10,20,22,55,47,71,68,67,21,54,56,52,58,61,23,19,25,26,45,60,70,65,57,50,49,59,69,72,24,51,40,53);(9,13,18,17)(10,21,25,49)(11,27,31,30)(12,32,36,35)(14,37,44,43)(15,38,42,46)(16,39,48,41)(19,50,53,67)(20,54,26,59)(22,56,45,69)(23,57,40,68)(24,47,58,70)(28,62,34,64)(29,63,66,33)(51,71,61,65)(52,60,72,55)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=i,e^2=gh,f^2=g,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234362, #72322387, #72322401
maximal normal subgroup of	#72234362, #72322387, #72322401

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420231);