#6596373

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×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2998 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6503569);