#8912478

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

4866 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

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, 8819674);