#7653759

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

4761 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=i,f^2=i,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, 7560955);