#7625123

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

1273 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)(19,27)(20,28)(21,31)(23,33)(25,34)(26,38)(29,39)(30,43)(32,42)(35,44)(36,48)(37,47)(40,41)(45,46);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,33)(28,34)(29,35)(30,36)(31,38)(32,37)(39,44)(40,45)(41,46)(42,47)(43,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,28,41,21,30,32,39,22,29,42,43,31,40,20,27)(18,23,34,46,26,36,37,44,24,35,47,48,38,45,25,33);(3,12)(7,10)(11,13)(15,16)(19,29)(23,35)(27,39)(30,40)(33,44)(36,45)(41,43)(46,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,24)(23,35)(25,37)(26,38)(33,44)(34,47)(36,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532319);