#9061395

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

4934 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)(13,15)(14,16)(17,18,21,25)(19,23,26,22)(20,24,29,32)(27,31,28,30);(1,2,5,6)(3,7,8,4)(9,11,13,16)(10,14,15,12)(19,26)(22,23)(27,28)(30,31);(1,2)(3,7)(4,8)(5,6)(10,15)(12,14)(17,19,21,26)(18,22,25,23)(20,28,29,27)(24,31,32,30);(2,6)(3,8)(9,10,13,15)(11,12,16,14)(17,19)(18,23)(20,28)(21,26)(22,25)(24,30)(27,29)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,20,21,29)(18,24,25,32)(19,27,26,28)(22,30,23,31);(1,4)(2,3)(5,7)(6,8)(9,12,13,14)(10,16,15,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968591);