#6584643

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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)(20,30)(22,29)(24,38)(25,33)(26,37)(27,34)(31,35)(32,40)(36,42)(39,43)(41,44);(1,2)(3,4)(6,10)(11,12)(13,15,18,27)(14,19,23,34)(16,24,31,41)(17,26,29,25)(20,32,36,43)(21,37,22,33)(28,38,35,44)(30,40,42,39);(2,4)(5,7,9,8)(6,11,10,12)(13,16,29,42,18,31,17,30)(14,20,21,35,23,36,22,28)(15,24,25,39,27,41,26,40)(19,32,37,44,34,43,33,38);(1,2)(3,4)(5,8,9,7)(6,12,10,11);(2,4)(13,17,18,29)(14,21,23,22)(15,25,27,26)(16,30,31,42)(19,33,34,37)(20,35,36,28)(24,39,41,40)(32,38,43,44)

Polycyclic presentation:

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

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, 6491839);