#9061564

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

4940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,e^2=g,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, 8968760);