#7623752

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
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

1523 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343987
maximal normal subgroup of	#72343987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530948);