#6497308

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469989
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1505 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404504);