#9685511

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

4720 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,16)(14,15)(17,18,23,26)(19,21,25,31)(20,22,30,24)(27,32,28,29);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,23,25)(18,21,26,31)(20,28,30,27)(22,29,24,32);(9,12)(10,15)(11,16)(13,14)(17,20,23,30)(18,24,26,22)(19,27,25,28)(21,29,31,32);(1,3,5,4)(2,6,7,8)(9,10,13,16)(11,14,15,12)(17,20)(18,22)(19,28)(21,29)(23,30)(24,26)(25,27)(31,32);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,14)(15,16)(17,21,23,31)(18,25,26,19)(20,29,30,32)(22,27,24,28);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,22)(18,20)(19,29)(21,28)(23,24)(25,32)(26,30)(27,31)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^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, 9592707);