#10181070

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

4884 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=hi,e^2=gi,f^2=g,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, 10088266);