#7625137

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

1279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgh,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, #72320093, #72320497
maximal normal subgroup of	#72287855, #72320093, #72320497

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532333);