#6438782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^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,g^c=gi

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, 6345978);