#7658351

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

4800 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565547);