#6584799

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1879 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)(7,11)(8,12)(9,10)(13,14)(15,19)(16,28)(17,21)(18,23)(20,32)(22,29)(24,39)(25,34)(26,35)(27,37)(30,38)(31,36)(33,40)(41,43)(42,44);(1,2)(3,4)(6,10)(11,12)(13,15)(14,19)(16,24)(17,27)(18,25)(20,33)(21,37)(22,35)(23,34)(26,29)(28,39)(30,41)(31,42)(32,40)(36,44)(38,43);(2,4)(5,7,9,8)(6,11,10,12)(13,16,29,31,18,30,17,32)(14,20,21,38,23,36,22,28)(15,24,26,42,25,41,27,40)(19,33,37,43,34,44,35,39);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,25)(16,30)(19,34)(20,36)(26,27)(28,38)(31,32)(35,37);(2,4)(13,17,18,29)(14,21,23,22)(15,26,25,27)(16,31,30,32)(19,35,34,37)(20,28,36,38)(24,40,41,42)(33,43,44,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491995);