#6516115

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	#25764	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,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, 6423311);