#8458451

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

4876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8365647);