#6519008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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

1435 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)(3,7)(4,6)(5,8)(17,18,22,26,21,23,34,37)(19,27,30,36,29,41,35,24)(20,31,33,47,32,25,39,40)(28,43,45,38,44,42,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,23)(24,36)(25,31)(26,37)(27,41)(38,46)(40,47)(42,43);(10,13)(11,15)(17,19,21,29)(18,24,23,36)(20,28,32,44)(22,35,34,30)(25,38,31,46)(26,41,37,27)(33,48,39,45)(40,43,47,42);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,32)(18,25,23,31)(19,28,29,44)(22,33,34,39)(24,38,36,46)(26,40,37,47)(27,42,41,43)(30,45,35,48);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(18,23)(20,32)(24,36)(26,37)(27,41)(28,44)(33,39)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,d^c=dh,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, 6426204);