#7624041

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

1324 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)(9,10)(11,21)(12,18)(13,19)(14,28)(15,31)(16,23)(17,24)(20,30)(22,35)(25,33)(26,38)(27,32)(29,36)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,23)(10,16,19,21)(12,25,15,22)(14,24,27,39)(17,28,37,32)(18,35,31,33)(20,34,38,36)(26,40,30,29);(9,11)(10,17)(12,22)(13,23)(14,29)(15,25)(16,32)(18,36)(19,37)(20,33)(21,28)(24,30)(26,39)(27,40)(31,34)(35,38);(2,5)(6,8)(9,12,13,15)(10,18,19,31)(11,22,23,25)(14,26,27,30)(16,33,21,35)(17,36,37,34)(20,28,38,32)(24,29,39,40);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,18,19,31)(11,22,23,25)(14,26,27,30)(16,33,21,35)(17,36,37,34)(20,28,38,32)(24,29,39,40)

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,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

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260392, #72322566
maximal normal subgroup of	#72260392, #72322566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531237);