#208562

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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=fi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dhi

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, 115758);