#7655222

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 7562418);