#9683956

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

4801 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9591152);