#7624673

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	#62474	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1228 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,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, 7531869);