#6638245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636836
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1210 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)(11,17,19,14)(12,20,22,15)(18,23,21,24)(25,26)(27,29)(28,35)(30,34)(31,37)(32,38)(33,39)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,29,26)(28,33,36,31)(30,34,38,32)(35,37,40,39);(3,7)(5,8)(25,28,30,37,29,36,38,39)(26,31,32,35,27,33,34,40);(3,7)(5,8)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(26,27)(31,33)(32,34)(35,40);(2,6)(5,8)(10,16)(14,17)(15,20)(23,24)(26,27)(31,33)(32,34)(35,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=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, 6545441);