#8096346

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003542);