#7625065

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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1199 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)(17,18,23,27)(19,24,22,37)(20,30,32,44)(21,35,36,48)(25,39,40,33)(26,29,41,31)(28,42,34,47)(38,46,45,43);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,23,22)(18,24,27,37)(20,28,32,34)(21,29,36,31)(25,38,40,45)(26,35,41,48)(30,42,44,47)(33,43,39,46);(17,20,31,46,22,34,36,39,23,32,29,43,19,28,21,33)(18,25,35,42,24,38,41,44,27,40,48,47,37,45,26,30);(3,10)(6,9)(11,16)(14,15);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(20,32)(25,40)(28,34)(30,44)(33,39)(38,45)(42,47)(43,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gh,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^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287855, #72320055, #72320482
maximal normal subgroup of	#72287855, #72320055, #72320482

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532261);