#224389

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131585);