#7624959

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624876
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	#33817	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1186 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)(19,24)(20,30)(21,34)(22,37)(23,27)(25,32)(26,35)(28,43)(29,42)(31,45)(33,46)(36,39)(38,44)(40,47)(41,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,22)(18,24,27,37)(20,28,33,31)(21,29,36,35)(25,38,41,40)(26,34,42,39)(30,43,46,45)(32,44,48,47);(17,20,29,44,22,31,21,32,23,33,35,47,19,28,36,48)(18,25,39,45,24,38,26,30,27,41,34,43,37,40,42,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,36,35)(25,40,41,38)(26,34,42,39)(30,45,46,43)(32,47,48,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=f,e^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=df,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, 7532155);