#9937839

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

5185 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)(10,14)(12,15)(17,18,22,26)(19,27,28,23)(20,24,25,21)(29,30,32,31);(1,2,4,6)(3,5,8,7)(9,10,13,14)(11,12,16,15)(19,28)(23,27)(29,32)(30,31);(9,10,13,14)(11,12,16,15)(17,19,22,28)(18,23,26,27)(20,29,25,32)(21,30,24,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,31)(25,32)(26,27);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,29)(21,26)(22,25)(23,31)(27,30)(28,32);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,14)(13,15)(17,21,22,24)(18,25,26,20)(19,30,28,31)(23,32,27,29)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,f^2=gh,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, 9845035);