#10155796

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

4834 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=g,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, 10062992);