#7643589

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

5278 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:

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

Polycyclic presentation:

2^9|b^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

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, 7550785);