#7625076

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1168 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532272);