#9888131

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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=g,e^2=g,f^2=i,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, 9795327);