#6585021

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=fi,e^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=dh,e^c=ei,e^d=eh

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, 6492217);