#7624874

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	#62332	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1291 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532070);