#10368467

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

4454 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=ghi,e^2=h,f^2=gh,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, 10275663);