#9684023

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

4946 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)(11,12)(13,16)(14,15)(17,18,23,25)(19,24,22,32)(20,26,29,21)(27,28,30,31);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,23,22)(18,24,25,32)(20,27,29,30)(21,31,26,28);(1,3,5,6)(2,4,7,8)(9,10,13,16)(11,14,15,12)(18,25)(20,29)(24,32)(27,30);(9,12)(10,15)(11,16)(13,14)(17,20)(18,26)(19,27)(21,25)(22,30)(23,29)(24,28)(31,32);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21,23,26)(18,20,25,29)(19,28,22,31)(24,30,32,27);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,22,23,19)(18,24,25,32)(20,30,29,27)(21,31,26,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,e^2=gh,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, 9591219);