#208552

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×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

898 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)(3,6)(4,5)(7,8)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22)(25,26,29,34,30,33,28,32)(27,31,35,38,36,39,37,40);(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23)(25,27,29,37,30,36,28,35)(26,31,32,38,33,39,34,40);(2,5)(6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)(25,28,30,29)(26,32,33,34)(27,35,36,37)(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=eg,c^2=h,d^2=f,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, 115748);