#9947047

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

5224 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)(4,6)(7,8)(10,12)(14,15)(19,25)(22,26)(27,31)(30,32);(1,2,4,6)(3,5,8,7)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,26,25,22)(20,23,29,28)(27,32,31,30);(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31);(1,3,4,8)(2,5,6,7)(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854243);