#7623433

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623323
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1220 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)(17,18,22,27)(19,24,28,38)(20,23,33,37)(21,34,32,26)(25,41,36,31)(29,35,42,46)(30,45,43,40)(39,48,47,44);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(20,31)(21,32)(24,35)(25,26)(27,37)(29,44)(30,43)(33,41)(34,36)(38,46)(39,40)(42,48)(45,47);(2,8)(6,9)(7,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,35)(25,39)(26,40)(27,38)(31,44)(32,43)(33,42)(34,45)(36,47)(37,46)(41,48);(19,28)(24,38)(29,42)(30,43)(35,46)(39,47)(40,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi,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, 7530629);