#6574791

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
prime radical	C₂×#3285	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2053 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)(13,14,18,19)(15,20,26,23)(16,21,28,32)(17,22,31,33)(24,34,38,37)(25,35,41,29)(27,36,42,30)(39,40,43,44);(6,9)(10,12)(13,15,18,26)(14,20,19,23)(16,24,28,38)(17,29,31,35)(21,34,32,37)(22,25,33,41)(27,40,42,44)(30,39,36,43);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(15,23)(16,21)(20,26)(22,33)(24,38)(25,29)(27,36)(28,32)(30,42)(35,41)(39,43);(13,16,18,28)(14,21,19,32)(15,24,26,38)(17,30,31,36)(20,34,23,37)(22,27,33,42)(25,40,41,44)(29,39,35,43);(13,17,18,31)(14,22,19,33)(15,25,26,41)(16,27,28,42)(20,35,23,29)(21,36,32,30)(24,39,38,43)(34,40,37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=i,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, 6481987);