#10506264

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

5013 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,12,16,14)(17,18,22,27)(19,25,26,21)(20,24,31,23)(28,30,29,32);(1,3,5,4)(2,6,7,8)(9,11,13,16)(10,14,15,12)(17,19)(18,21)(20,28)(22,26)(23,30)(24,32)(25,27)(29,31);(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,28)(21,30)(22,31)(24,27)(25,32)(26,29);(1,2)(3,8)(4,6)(5,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,29)(21,30)(22,31)(23,27)(25,32)(26,28);(1,4,5,3)(2,6,7,8)(17,19)(18,25)(20,28)(21,27)(22,26)(23,32)(24,30)(29,31);(1,2)(3,6)(4,8)(5,7)(17,21,22,25)(18,26,27,19)(20,30,31,32)(23,29,24,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10413460);