#6378890

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1247 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)(11,18)(12,22)(13,16)(14,17)(15,19)(20,23)(21,24)(25,26,30,32,29,33,28,37)(27,34,35,39,36,31,38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,18)(22,24)(25,27,29,36)(26,31,33,34)(28,38,30,35)(32,40,37,39);(2,6)(5,8)(11,19)(15,18)(20,21)(23,24)(25,27)(26,31)(28,38)(29,36)(30,35)(32,40)(33,34)(37,39);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,23,18,24)(25,28,29,30)(26,32,33,37)(27,35,36,38)(31,39,34,40);(9,13,14,12)(10,16,17,22)(11,21,19,20)(15,23,18,24)(25,26,28,37,29,33,30,32)(27,34,38,40,36,31,35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286086);