#7653435

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

5737 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:

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

Polycyclic presentation:

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

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, 7560631);