#7654730

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

4812 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)(10,12)(11,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,32)(30,31);(1,2,5,6)(3,7,4,8)(9,10,13,12)(11,16,15,14)(17,19,22,29)(18,23,26,27)(20,21,28,31)(24,30,32,25);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,14,12,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,32)(30,31);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,16,15,14)(17,20,22,28)(18,24,26,32)(19,21,29,31)(23,30,27,25);(1,3,5,4)(2,7,6,8)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,29)(28,32);(1,4,5,3)(2,8,6,7)(17,21)(18,25)(19,28)(20,29)(22,31)(23,24)(26,30)(27,32)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=g,e^2=gi,f^2=gi,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, 7561926);