#9848580

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

4829 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)(17,18,22,26)(19,23,21,25)(20,24,29,28)(27,31,30,32);(2,3)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,29)(27,32)(30,31);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,31,28,32);(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,15)(14,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,32)(26,28);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21,22,19)(18,25,26,23)(20,27,29,30)(24,31,28,32)

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,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, 9755776);