#9937752

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

4777 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)(3,6)(4,5)(7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,26)(21,29)(22,24)(25,28)(27,32)(30,31);(2,5)(6,7)(10,14)(11,15)(17,19,22,28)(18,23,24,25)(20,29,31,27)(21,30,32,26);(9,11)(10,12)(13,15)(14,16)(18,24)(19,28)(26,30)(27,29);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,32)(27,31);(1,2)(3,7)(4,5)(6,8)(17,20)(18,26)(19,29)(21,23)(22,31)(24,30)(25,32)(27,28);(1,3)(2,6)(4,8)(5,7)(17,21,22,32)(18,27,24,29)(19,30,28,26)(20,25,31,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,f^2=h,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, 9844948);