#513040

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	C₂²×D₁₆	C₂²
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

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

normal subgroups only

This group has 398 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=e,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, 420236);