#213574

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1129 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120770);