#216839

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

816 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)(3,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(25,26,30,34,31,28,29,33)(27,32,36,39,38,35,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,27,31,38)(26,32,28,35)(29,37,30,36)(33,40,34,39);(2,5)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(25,28,31,26)(27,35,38,32)(29,34,30,33)(36,40,37,39);(1,3,4,7)(2,6,5,8)(25,29,31,30)(26,33,28,34)(27,36,38,37)(32,39,35,40)

Polycyclic presentation:

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

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, 124035);