#7623842

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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1283 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:

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

Polycyclic presentation:

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

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, 7531038);