#6584311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491507);