#9936475

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

4916 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=h,e^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, 9843671);