#6502308

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(3,8)(4,6)(5,7)(17,18,22,28)(19,29,33,24)(20,26,35,44)(21,27,23,38)(25,37,45,34)(30,40,31,46)(32,42,36,41)(39,48,43,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,29)(20,31,35,30)(21,36,23,32)(25,43,45,39)(26,40,44,46)(27,42,38,41)(34,47,37,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,35)(18,25,28,45)(19,30,33,31)(21,34,23,37)(24,39,29,43)(26,27,44,38)(32,47,36,48)(40,41,46,42);(11,16)(13,15)(17,20)(18,26)(19,31)(21,37)(22,35)(23,34)(24,40)(25,38)(27,45)(28,44)(29,46)(30,33)(32,47)(36,48)(39,42)(41,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,22,23)(18,27,28,38)(19,32,33,36)(20,34,35,37)(24,41,29,42)(25,44,45,26)(30,47,31,48)(39,46,43,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6409504);