#9848590

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

4776 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:

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

Polycyclic presentation:

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

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, 9755786);