#7645335

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

4955 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7552531);