#222932

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
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×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

969 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)(6,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,30,34,29,32,28,33)(27,35,37,40,36,31,38,39);(1,2)(3,6)(4,5)(7,8)(25,27,29,36)(26,31,32,35)(28,38,30,37)(33,40,34,39);(1,3,4,7)(2,6,5,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(26,32)(27,36)(28,30)(39,40);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,29,30)(26,33,32,34)(27,37,36,38)(31,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130128);