#7624056

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

1290 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)(14,26)(17,31)(19,36)(21,39)(22,38)(24,25)(27,28)(29,33)(30,35)(32,40)(34,37);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,22,29)(12,23,15,18)(14,19,25,30)(20,33,38,31)(21,34,28,32)(24,36,26,35)(27,37,39,40);(9,11)(10,17)(12,21)(13,22)(14,27)(15,28)(16,29)(18,32)(19,37)(20,26)(23,34)(24,38)(25,39)(30,40)(31,36)(33,35);(2,5)(6,8)(9,12,13,15)(10,18,16,23)(11,21,22,28)(14,24,25,26)(17,32,29,34)(19,35,30,36)(20,27,38,39)(31,37,33,40);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,18,16,23)(11,21,22,28)(14,24,25,26)(17,32,29,34)(19,35,30,36)(20,27,38,39)(31,37,33,40)

Polycyclic presentation:

2^9|b^2=i,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,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	#72322060
maximal normal subgroup of	#72322060
maximal quotient of	#72343811, #72343816

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531252);