#7625146

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

1274 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,27)(19,23,30,26)(20,31,34,47)(21,35,36,38)(24,39,41,33)(25,42,43,29)(28,44,32,46)(37,48,40,45);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,30)(18,23,27,26)(20,28,34,32)(21,29,36,42)(24,37,41,40)(25,38,43,35)(31,44,47,46)(33,45,39,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,29,45,19,28,36,39,22,34,42,48,30,32,21,33)(18,24,35,46,26,40,43,47,27,41,38,44,23,37,25,31);(3,12)(7,10)(11,13)(15,16)(17,19,22,30)(18,23,27,26)(20,28,34,32)(21,29,36,42)(24,37,41,40)(25,38,43,35)(31,44,47,46)(33,45,39,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,29,36,42)(24,40,41,37)(25,38,43,35)(31,46,47,44)(33,48,39,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=i,e^2=fi,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, #72320683
maximal normal subgroup of	#72287855, #72320093, #72320683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532342);