#10216612

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

4656 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=hi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10123808);