#9947336

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

4850 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=ghi,e^2=g,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, 9854532);