#208187

order	512
nilpotency class	3
isoclinism	isoclinic class of #161458
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

922 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)(3,6)(4,5)(7,8)(9,10,12,14,13,15,11,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(10,14)(11,12)(15,16)(17,19,20,27,22,29,21,28)(18,23,30,32,25,31,24,26);(2,5)(6,8)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32);(1,3,4,7)(2,6,5,8)(10,15)(14,16)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=f,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi

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, 115383);