#9726964

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

4953 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 9634160);