#7624157

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,18)(11,20,22,39)(12,17,15,24)(14,27,26,37)(16,31,33,23)(19,25,35,29)(21,38,30,36)(28,32,40,34);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,24)(13,18)(14,19)(15,17)(20,23)(21,34)(22,33)(25,37)(26,35)(27,29)(28,36)(30,32)(31,39)(38,40);(9,11,13,22)(10,16,18,33)(12,21,15,30)(14,28,26,40)(17,32,24,34)(19,36,35,38)(20,37,39,27)(23,25,31,29);(3,8)(5,7)(9,12,13,15)(10,17,18,24)(11,21,22,30)(14,25,26,29)(16,32,33,34)(19,27,35,37)(20,38,39,36)(23,28,31,40);(2,6)(5,7)(10,18)(16,33)(17,24)(19,35)(20,39)(27,37)(32,34)(36,38)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343273
maximal normal subgroup of	#72343273
maximal quotient of	#72343840, #72343842, #72343843, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531353);