#7650532

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

4733 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:

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

Polycyclic presentation:

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

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, 7557728);