#217167

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

839 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)(11,17)(12,20,22,15)(13,16)(14,19)(18,23,21,24)(25,26,29,32,28,31,36,39)(27,33,35,40,34,30,37,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,37)(39,40);(2,5)(6,8)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(27,34)(30,33)(35,37)(38,40);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,20)(18,21)(26,31)(27,34)(32,39)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=ehi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,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, 124363);