#7623846

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,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,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260392
maximal normal subgroup of	#72260392

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531042);