#7624066

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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,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	#72313684, #72315871, #72315916, …
maximal normal subgroup of	#72313684, #72315871, #72315916, …
maximal quotient of	#72343810, #72343815

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531262);