#6444278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2700 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)(12,23,25,33)(13,18,14,28)(17,27,34,24)(20,35,38,40)(21,30,26,31)(29,36,39,37);(1,2)(3,4)(6,7)(9,11)(10,16)(12,20)(13,26)(14,21)(15,22)(17,29)(18,31)(19,32)(23,35)(24,37)(25,38)(27,36)(28,30)(33,40)(34,39);(2,4)(5,6,8,7)(9,12,13,27,15,25,14,24)(10,17,28,33,19,34,18,23)(11,20,26,36,22,38,21,37)(16,29,30,40,32,39,31,35);(2,4)(5,7,8,6)(9,11)(10,16)(12,20)(13,26)(14,21)(15,22)(17,29)(18,31)(19,32)(23,35)(24,37)(25,38)(27,36)(28,30)(33,40)(34,39);(9,13,15,14)(10,18,19,28)(11,21,22,26)(12,24,25,27)(16,30,32,31)(17,33,34,23)(20,36,38,37)(29,35,39,40)

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,e^c=ei,e^d=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, 6351474);