#7658951

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

4720 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566147);