#7651247

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

4842 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)(10,13)(11,15)(17,18)(19,21)(20,25)(22,24)(23,27)(26,29)(28,31)(30,32);(1,2,5,6)(3,7,4,8)(10,13)(14,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,30)(29,32);(1,3,5,4)(2,7,6,8)(9,10,12,13)(11,14,15,16)(18,24)(21,23)(25,32)(29,31);(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32);(17,20,22,30)(18,25,24,32)(19,26,27,28)(21,29,23,31);(1,4,5,3)(2,8,6,7)(17,21,22,23)(18,19,24,27)(20,29,30,31)(25,26,32,28)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7558443);