#7624954

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	#33737	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)(21,35)(22,37)(23,27)(25,33)(26,31)(28,41)(29,40)(32,43)(34,47)(36,48)(38,42)(39,46)(44,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,22)(18,24,27,37)(20,28,32,34)(21,29,36,31)(25,38,39,44)(26,35,40,48)(30,41,43,47)(33,42,46,45);(17,20,31,45,22,34,36,46,23,32,29,42,19,28,21,33)(18,25,35,41,24,38,40,43,27,39,48,47,37,44,26,30);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(20,32)(25,39)(28,34)(30,43)(33,46)(38,44)(41,47)(42,45)

Polycyclic presentation:

2^9|b^2=fi,c^2=gh,d^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^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, 7532150);