#6605074

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1408 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)(19,29,31,24)(20,34,36,42)(21,28,23,27)(25,37,41,35)(30,44,45,48)(32,39,33,40)(38,46,43,47);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,31)(18,24,26,29)(20,30,36,45)(21,32,23,33)(25,38,41,43)(27,39,28,40)(34,48,42,44)(35,46,37,47);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,46)(33,47)(39,48)(40,44);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(18,26)(19,31)(25,41)(27,28)(30,45)(32,33)(34,42)(46,47);(1,4,5,3)(2,6,8,7)(10,13)(11,15)(17,21,22,23)(18,27,26,28)(19,32,31,33)(20,35,36,37)(24,39,29,40)(25,34,41,42)(30,46,45,47)(38,48,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512270);