#7624893

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1330 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)(10,17)(11,20)(13,16)(15,29)(18,32)(19,33)(21,28)(22,36)(23,39)(24,38)(26,31)(27,37)(30,35)(34,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,16)(13,25)(14,17)(15,19)(20,35)(21,33)(22,23)(24,30)(26,29)(27,34)(28,31)(32,38)(36,37)(39,40);(9,11,21,39,16,23,15,20,14,24,29,37,13,27,28,38)(10,18,33,40,12,22,19,35,17,30,26,36,25,34,31,32);(3,8)(5,7)(9,12)(10,13)(11,22)(14,25)(15,26)(16,17)(18,27)(19,21)(20,36)(23,30)(24,34)(28,33)(29,31)(32,37)(35,39)(38,40);(2,6)(5,7)(9,13,14,16)(10,12,17,25)(11,23,24,27)(15,21,28,29)(18,34,30,22)(19,26,31,33)(20,37,38,39)(32,36,35,40)

Polycyclic presentation:

2^9|c^2=ghi,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,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315178, #72315814, #72315871
maximal normal subgroup of	#72315178, #72315814, #72315871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532089);