#6443273

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

4932 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)(9,10,16,22)(11,23,27,34)(12,28,30,36)(13,19,33,40)(14,20,15,21)(17,26,35,25)(18,29,37,24)(31,38,32,39);(1,2,3,4)(5,6)(7,8)(9,11,14,25,16,27,15,26)(10,17,21,23,22,35,20,34)(12,13,29,32,30,33,24,31)(18,19,36,38,37,40,28,39);(6,8)(9,12,14,29,16,30,15,24)(10,18,21,36,22,37,20,28)(11,13,25,32,27,33,26,31)(17,19,23,38,35,40,34,39);(6,8)(9,13)(10,19)(11,24)(12,25)(14,32)(15,31)(16,33)(17,28)(18,23)(20,39)(21,38)(22,40)(26,30)(27,29)(34,37)(35,36);(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,24,30,29)(13,31,33,32)(17,34,35,23)(18,36,37,28)(19,38,40,39)

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=dhi,e^c=ei,e^d=ei

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, 6350469);