#7624351

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,21)(19,28,27,24)(20,31,33,46)(23,38,37,30)(25,42,32,34)(26,43,36,40)(29,39,45,47)(35,48,41,44);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,41)(26,40)(31,47)(32,35)(33,45)(34,44)(36,43)(37,38)(39,46)(42,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,39)(24,41)(26,44)(27,45)(28,35)(30,46)(31,38)(34,40)(36,48)(37,47)(42,43);(3,11)(6,9)(10,16)(13,15)(17,21,22,18)(19,28,27,24)(20,32,33,25)(23,36,37,26)(29,35,45,41)(30,43,38,40)(31,34,46,42)(39,48,47,44);(2,8)(6,9)(7,14)(13,15)(18,21)(19,27)(25,32)(26,36)(29,45)(30,38)(31,46)(44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^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=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531547);