#6465729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3174 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6372925);