#6548546

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=ei,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, 6455742);