#7642666

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

4996 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)(19,26)(20,24)(21,22)(23,25)(27,32)(28,31)(29,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,15,12)(17,19,23,20)(18,24,25,26)(21,28,27,30)(22,29,32,31);(9,11)(10,12)(13,16)(14,15)(18,25)(21,27)(24,26)(28,30);(9,12)(10,11)(13,14)(15,16)(17,20,23,19)(18,26,25,24)(21,30,27,28)(22,31,32,29);(1,3,4,7)(2,6,5,8)(17,21,23,27)(18,22,25,32)(19,28,20,30)(24,29,26,31);(17,22)(18,27)(19,29)(20,31)(21,25)(23,32)(24,30)(26,28)

Polycyclic presentation:

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

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, 7549862);