#10320939

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

5135 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)(11,16)(12,15)(13,14)(17,18,22,21)(19,25,26,23)(20,24,28,30)(27,32,29,31);(1,2)(3,7)(4,5)(6,8)(9,11,14,12)(10,15,13,16)(18,21)(19,26)(24,30)(27,29);(2,5)(6,7)(9,10,14,13)(11,15,12,16)(20,28)(24,30)(27,29)(31,32);(2,5)(6,7)(9,10,14,13)(11,16,12,15)(17,19,22,26)(18,23,21,25)(20,29,28,27)(24,32,30,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,20,22,28)(18,24,21,30)(19,27,26,29)(23,31,25,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,15,12,16)(17,21,22,18)(19,25,26,23)(20,30,28,24)(27,32,29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10228135);