#6545582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

1586 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)(17,18,23,26,22,28,21,27)(19,24,31,41,30,38,32,39)(20,33,34,25,36,42,35,40)(29,43,46,37,44,48,45,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,32)(22,30)(23,31)(25,37)(26,41)(27,39)(28,38)(33,43)(34,46)(35,45)(36,44)(40,47)(42,48);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,33)(27,42)(28,40)(30,44)(31,45)(32,46)(38,47)(39,48)(41,43);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,38)(29,44)(31,32)(37,47)(39,41)(43,48)(45,46);(11,15)(14,16)(17,21,22,23)(18,26,28,27)(19,31,30,32)(20,34,36,35)(24,39,38,41)(25,33,40,42)(29,45,44,46)(37,48,47,43)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452778);