#8912304

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

5383 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 8819500);