#6583171

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3076 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)(9,10,15,19)(11,20,23,30)(12,17,25,33)(13,18,14,28)(16,26,31,22)(21,35,38,40)(24,32,27,34)(29,36,39,37);(5,6)(7,8)(9,11,13,26,15,23,14,22)(10,16,28,30,19,31,18,20)(12,21,27,36,25,38,24,37)(17,29,32,40,33,39,34,35);(1,2,3,4)(6,8)(9,12,15,25)(10,17,19,33)(11,21,23,38)(13,27,14,24)(16,29,31,39)(18,34,28,32)(20,35,30,40)(22,37,26,36);(5,6)(7,8)(9,13,15,14)(10,18,19,28)(11,22,23,26)(12,24,25,27)(16,30,31,20)(17,32,33,34)(21,36,38,37)(29,35,39,40);(6,8)(11,23)(12,25)(16,31)(17,33)(20,30)(22,26)(24,27)(32,34)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

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, 6490367);