#7625134

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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

1269 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)(17,18,21,25)(19,28,31,45)(20,24,34,27)(22,36,35,41)(23,37,39,32)(26,43,42,29)(30,44,33,48)(38,46,40,47);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,18)(19,23)(20,27)(21,25)(22,26)(24,34)(28,32)(29,36)(30,40)(31,39)(33,38)(35,42)(37,45)(41,43)(44,46)(47,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,29,46,20,33,35,37,21,31,43,47,34,30,22,32)(18,23,36,44,27,38,42,45,25,39,41,48,24,40,26,28);(3,12)(7,10)(11,13)(15,16)(17,20,21,34)(18,24,25,27)(19,30,31,33)(22,29,35,43)(23,38,39,40)(26,41,42,36)(28,44,45,48)(32,47,37,46);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,25)(23,39)(24,27)(26,42)(28,45)(36,41)(38,40)(44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318881, #72320093, #72320497
maximal normal subgroup of	#72318881, #72320093, #72320497

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532330);