#7624164

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

1555 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)(11,17)(12,24)(13,19)(14,28)(15,32)(16,21)(18,26)(20,30)(22,33)(23,37)(25,35)(27,38)(29,36)(31,40)(34,39);(1,3)(2,5)(4,8)(6,7)(9,11,16,12)(10,17,21,24)(13,25,15,22)(14,23,31,26)(18,28,37,40)(19,35,32,33)(20,34,38,36)(27,29,30,39);(9,12,16,11)(10,18,21,37)(13,25,15,22)(14,29,31,39)(17,28,24,40)(19,36,32,34)(20,33,38,35)(23,30,26,27);(3,8)(5,7);(2,6)(5,7)(9,13,16,15)(10,19,21,32)(11,22,12,25)(14,27,31,30)(17,33,24,35)(18,36,37,34)(20,28,38,40)(23,39,26,29)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^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	#72318797, #72320368, #72320623
maximal normal subgroup of	#72318797, #72320368, #72320623

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531360);