#10366273

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

4658 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)(17,18,22,23)(19,20,25,24)(21,29,30,26)(27,31,32,28);(9,10,13,15)(11,12,16,14)(18,23)(19,25)(21,30)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,15,12)(18,23)(20,24)(26,29)(28,31);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=gi,e^2=h,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, 10273469);