#7625177

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1266 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)(19,24)(20,33)(21,25)(22,35)(23,27)(26,37)(28,42)(29,39)(30,40)(31,44)(32,41)(34,38)(36,48)(43,47)(45,46);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,23,32)(18,24,27,41)(20,28,36,45)(21,29,37,30)(22,31,38,47)(25,39,26,40)(33,42,48,46)(34,43,35,44);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,33,18,21,35,38,26,23,36,48,27,37,34,22,25)(19,28,42,24,29,44,47,40,32,45,46,41,30,43,31,39);(3,12)(7,10)(11,13)(15,16)(17,21,23,37)(18,25,27,26)(19,29,32,30)(20,34,36,35)(22,33,38,48)(24,39,41,40)(28,43,45,44)(31,42,47,46);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,21,23,37)(18,25,27,26)(19,30,32,29)(20,34,36,35)(22,33,38,48)(24,40,41,39)(28,44,45,43)(31,46,47,42)

Polycyclic presentation:

2^9|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^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287059, #72320093, #72320368
maximal normal subgroup of	#72287059, #72320093, #72320368

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532373);