#6443231

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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
prime radical	C₂³×#135	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5054 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:

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

Polycyclic presentation:

2^9|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^b=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, 6350427);