#6444238

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,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=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351434);