#6552670

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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)(13,14,17,22,18,21,32,38)(15,23,27,34,25,19,26,35)(16,28,31,36,30,37,29,20)(24,40,42,33,41,44,43,39);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,27)(18,25)(20,33)(21,23)(22,35)(26,32)(28,44)(29,43)(30,41)(31,42)(34,38)(36,39)(37,40);(1,3,4,2)(6,9)(10,12)(13,16)(14,20)(15,24)(17,29)(18,30)(19,33)(21,36)(22,37)(23,39)(25,41)(26,42)(27,43)(28,38)(31,32)(34,44)(35,40);(5,7)(6,10)(8,11)(9,12)(14,21)(15,25)(20,36)(22,38)(24,41)(26,27)(28,37)(42,43);(7,11)(10,12)(13,17,18,32)(14,22,21,38)(15,26,25,27)(16,29,30,31)(19,34,23,35)(20,37,36,28)(24,42,41,43)(33,44,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459866);