#9789148

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
prime radical	#47840	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4619 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9696344);