#6574863

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2029 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);(6,9)(10,12)(13,14,18,23)(15,24,28,20)(16,29,32,36)(17,19,33,22)(21,31,39,34)(25,40,27,35)(26,38,42,44)(30,37,43,41);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(15,25)(17,33)(21,36)(22,23)(26,30)(27,28)(29,39)(31,34)(35,40)(38,44)(42,43);(13,15,18,28)(14,20,23,24)(16,30,32,43)(17,25,33,27)(19,35,22,40)(21,38,39,44)(26,34,42,31)(29,41,36,37);(13,16,18,32)(14,21,23,39)(15,26,28,42)(17,34,33,31)(19,36,22,29)(20,37,24,41)(25,30,27,43)(35,44,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482059);