#7656913

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

5135 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564109);