#6552576

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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=i,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, 6459772);