#6465577

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3186 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)(9,10,13,17,14,18,27,34)(11,19,22,28,20,33,26,15)(12,16,25,32,24,31,39,40)(21,35,36,30,23,37,38,29);(6,7)(9,11,14,20)(10,15,18,28)(12,23,24,21)(13,26,27,22)(16,30,31,29)(17,33,34,19)(25,36,39,38)(32,35,40,37);(1,2,3,4)(5,6,8,7)(11,20)(15,28)(19,33)(21,23)(22,26)(29,30)(35,37)(36,38);(5,7,8,6)(9,12,14,24)(10,16,18,31)(11,21,20,23)(13,25,27,39)(15,29,28,30)(17,32,34,40)(19,35,33,37)(22,36,26,38);(12,24)(16,31)(21,23)(25,39)(29,30)(32,40)(35,37)(36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372773);