#6575154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2193 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,26,36)(16,20,30,40)(17,21,32,42)(23,33,43,44)(24,34,25,35)(27,37,28,38)(29,39,31,41);(6,9)(10,12)(13,15)(14,19)(16,27)(17,24)(18,26)(20,37)(21,34)(22,36)(23,29)(25,32)(28,30)(31,43)(33,39)(35,42)(38,40)(41,44);(2,4)(13,16,18,30)(14,20,22,40)(15,23,26,43)(17,31,32,29)(19,33,36,44)(21,41,42,39)(24,27,25,28)(34,37,35,38);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,28)(17,24)(18,26)(20,38)(21,34)(22,36)(23,31)(25,32)(27,30)(29,43)(33,41)(35,42)(37,40)(39,44);(15,24)(17,32)(19,34)(21,42)(23,27)(25,26)(28,43)(29,31)(33,37)(35,36)(38,44)(39,41)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6482350);