#7624968

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 ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1184 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)(19,29)(20,34)(21,26)(22,38)(23,39)(24,31)(25,28)(27,35)(30,44)(32,45)(33,42)(36,37)(40,43)(41,47)(46,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,31)(18,24,26,29)(20,30,27,41)(22,32,37,46)(23,33,25,40)(28,42,39,43)(34,47,35,44)(36,45,38,48);(17,20,35,26,23,36,37,39,21,27,34,18,25,38,22,28)(19,30,44,29,33,45,46,43,31,41,47,24,40,48,32,42);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,31)(24,29)(30,41)(32,46)(33,40)(42,43)(44,47)(45,48);(2,8)(6,9)(7,16)(14,15)(18,26)(19,31)(20,27)(28,39)(32,46)(33,40)(36,38)(44,47)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=fi,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, 7532164);