#6516736

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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)(3,7,4,6)(17,18)(19,28)(20,26)(21,27)(22,24)(23,36)(25,30)(29,45)(31,41)(32,39)(33,48)(34,43)(35,44)(37,38)(40,47)(42,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,24)(20,32)(25,38)(27,36)(28,41)(29,33)(34,35)(46,47);(10,13)(11,15)(17,19,22,31)(18,25,24,38)(20,33,32,29)(21,30,23,37)(26,42,39,40)(27,41,36,28)(34,47,35,46)(43,48,44,45);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,26,24,39)(19,29,31,33)(21,35,23,34)(25,40,38,42)(27,44,36,43)(28,45,41,48)(30,47,37,46);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(17,21,22,23)(18,27,24,36)(19,30,31,37)(20,34,32,35)(25,41,38,28)(26,43,39,44)(29,46,33,47)(40,45,42,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi,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, 6423932);