#7655252

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	#39958	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562448);