#6596222

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2982 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6503418);