#9780531

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	#59122	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4711 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,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, 9687727);