#9063297

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

4845 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)(3,5)(4,8)(6,7)(9,10,13,11)(12,15,16,14)(17,18,23,26)(19,24,27,20)(21,22,25,32)(28,31,30,29);(2,7)(3,8)(10,11)(14,15)(17,19)(18,24)(20,26)(21,30)(22,29)(23,27)(25,28)(31,32);(1,3,6,8)(2,4,7,5)(9,11,13,10)(12,15,16,14)(19,27)(20,24)(21,25)(22,32);(1,4,6,5)(2,3,7,8)(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,31)(22,30)(23,24)(25,29)(26,27)(28,32);(1,2)(3,4)(5,8)(6,7)(17,21,23,25)(18,22,26,32)(19,28,27,30)(20,29,24,31);(1,5,6,4)(2,8,7,3)(17,22)(18,25)(19,29)(20,30)(21,26)(23,32)(24,28)(27,31)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=g,e^2=gi,f^2=h,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, 8970493);