#7602641

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,32)(21,35)(22,27)(23,39)(24,30)(26,38)(28,33)(29,45)(31,41)(34,40)(36,42)(37,48)(43,47)(44,46);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,25)(20,33,31,23)(21,36,37,34)(26,43,44,45)(28,32,39,41)(29,46,47,38)(35,40,48,42);(1,3,4,7)(2,6,5,8)(17,20,33,30,22,31,23,19)(18,25,39,41,27,24,28,32)(21,36,46,29,37,34,38,47)(26,40,48,45,44,42,35,43);(11,16)(13,15)(17,21,23,38,22,37,33,46)(18,26,28,35,27,44,39,48)(19,29,31,36,30,47,20,34)(24,40,41,43,25,42,32,45);(2,5)(6,8)(10,14)(13,15)(18,27)(21,37)(24,25)(28,39)(29,47)(32,41)(34,36)(38,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fh,d^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7509837);