#7624280

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624128
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
prime radical	C₂×#3341	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1268 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,28)(19,29,30,46)(20,27,22,25)(21,34,37,42)(24,39,40,32)(26,33,44,36)(31,47,38,43)(35,45,48,41);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(19,30)(24,40)(29,46)(31,38)(32,39)(35,48)(41,45)(43,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,24)(20,31)(21,35)(22,38)(23,30)(25,41)(26,43)(27,45)(28,40)(29,34)(32,36)(33,39)(37,48)(42,46)(44,47);(3,11)(6,9)(10,16)(13,15)(17,20,23,22)(18,25,28,27)(19,31,30,38)(21,33,37,36)(24,41,40,45)(26,42,44,34)(29,43,46,47)(32,35,39,48);(2,8)(6,9)(7,14)(13,15)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531476);