#7624830

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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
prime radical	#62270	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1333 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)(10,16)(11,20)(12,15)(14,27)(17,30)(19,31)(21,26)(22,37)(23,36)(24,35)(28,33)(29,34)(32,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,25)(13,16)(14,19)(15,18)(20,33)(21,31)(22,38)(23,28)(24,32)(26,29)(27,34)(30,36)(35,39)(37,40);(9,11,21,35,15,24,14,20,13,23,27,37,12,22,26,36)(10,17,31,39,18,32,19,33,16,28,34,40,25,38,29,30);(3,8)(5,7)(10,16)(17,28)(18,25)(19,29)(30,33)(31,34)(32,38)(39,40);(2,6)(5,7)(9,12,13,15)(10,18,16,25)(11,22,23,24)(14,21,26,27)(17,32,28,38)(19,34,29,31)(20,35,36,37)(30,39,33,40)

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314254, #72315824, #72315846
maximal normal subgroup of	#72314254, #72315824, #72315846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532026);