#513042

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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)(14,41)(15,45)(16,47)(17,49)(18,25)(20,31)(22,27)(23,44)(24,42)(28,57)(29,56)(30,64)(32,50)(33,62)(34,61)(35,54)(36,55)(37,53)(38,68)(39,69)(40,60)(43,59)(46,58)(48,70)(51,67)(52,66)(63,72)(65,71);(2,5)(6,8)(9,11,27,34,16,30,38,37,13,36,46,63,40,67,44,35,18,31,15,29,48,65,43,32,17,28,39,66,42,33,14,12)(10,19,41,62,24,52,69,57,49,50,59,71,70,56,45,20,25,54,23,51,60,72,58,55,21,53,68,64,47,61,22,26);(1,3,4,7)(2,6,5,8)(9,12,14,33,42,66,39,28,17,32,43,65,48,29,15,31,18,35,44,67,40,63,46,36,13,37,38,30,16,34,27,11)(10,20,22,56,47,71,68,50,21,57,58,52,60,62,23,19,25,26,45,61,70,64,59,53,49,55,69,72,24,51,41,54);(9,13,18,17)(10,21,25,49)(11,28,31,36)(12,32,35,37)(14,38,44,43)(15,39,27,46)(16,40,48,42)(19,50,54,53)(20,55,26,57)(22,58,45,69)(23,59,41,68)(24,47,60,70)(29,63,34,66)(30,33,65,67)(51,64,62,71)(52,56,72,61)

Polycyclic presentation:

2^9|b^2=fi,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,d^b=di,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420238);