#6583130

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3098 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)(11,20,24,34)(12,17,26,32)(13,18,14,28)(16,22,30,23)(21,33,36,40)(25,31,27,38)(29,35,39,37);(5,6)(7,8)(9,11,14,23,15,24,13,22)(10,16,18,20,19,30,28,34)(12,21,27,37,26,36,25,35)(17,29,31,33,32,39,38,40);(1,2,3,4)(6,8)(9,12)(10,17)(11,21)(13,25)(14,27)(15,26)(16,29)(18,31)(19,32)(20,33)(22,35)(23,37)(24,36)(28,38)(30,39)(34,40);(5,6)(7,8)(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,25,26,27)(16,20,30,34)(17,31,32,38)(21,35,36,37)(29,33,39,40);(6,8)(12,26)(17,32)(21,36)(25,27)(29,39)(31,38)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,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, 6490326);