#6552566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459762);