#6443129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4976 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)(9,10,14,18)(11,19,23,36)(12,16,24,32)(13,26,27,17)(15,28,31,22)(20,34,21,35)(25,40,38,33)(29,39,30,37);(5,6)(7,8)(9,11,13,22,14,23,27,28)(10,15,17,19,18,31,26,36)(12,20,25,37,24,21,38,39)(16,29,33,34,32,30,40,35);(1,2,3,4)(6,8)(9,12)(10,16)(11,20)(13,25)(14,24)(15,29)(17,33)(18,32)(19,34)(21,23)(22,37)(26,40)(27,38)(28,39)(30,31)(35,36);(6,8)(9,12)(10,16)(11,21)(13,25)(14,24)(15,30)(17,33)(18,32)(19,35)(20,23)(22,39)(26,40)(27,38)(28,37)(29,31)(34,36);(12,24)(16,32)(20,21)(25,38)(29,30)(33,40)(34,35)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6350325);