#6584259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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:

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

Polycyclic presentation:

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

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, 6491455);