#206879

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,e^b=eh,g^b=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114075);