#6438787

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3197 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)(9,10,15,20)(11,16,23,31)(12,25,27,34)(13,19,14,18)(17,28,33,26)(21,35,37,40)(22,32,24,30)(29,38,39,36);(1,2,4,3)(5,6)(7,8)(9,11)(10,16)(12,21)(13,22)(14,24)(15,23)(17,29)(18,30)(19,32)(20,31)(25,35)(26,36)(27,37)(28,38)(33,39)(34,40);(1,3,4,2)(6,8)(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,25)(19,34)(20,33)(22,36)(23,37)(24,38)(30,35)(31,39)(32,40);(6,8)(9,13,15,14)(10,18,20,19)(11,22,23,24)(12,26,27,28)(16,30,31,32)(17,25,33,34)(21,36,37,38)(29,35,39,40);(11,23)(16,31)(21,37)(22,24)(29,39)(30,32)(35,40)(36,38)

Polycyclic presentation:

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

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, 6345983);