#6443266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

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

4946 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6350462);