#7929459

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

5083 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)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,23)(21,26)(22,25)(27,30)(28,31)(29,32);(1,2)(3,7)(4,8)(5,6)(17,19,23,26)(18,24,20,21)(22,27,31,32)(25,30,28,29);(10,14)(11,15)(18,20)(19,26)(25,28)(27,32);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,12,14,16)(17,20,23,18)(19,21,26,24)(22,28,31,25)(27,29,32,30);(9,12)(10,15)(11,14)(13,16)(17,21)(18,19)(20,26)(22,30)(23,24)(25,32)(27,28)(29,31);(1,4,5,3)(2,8,6,7)(17,22,23,31)(18,25,20,28)(19,27,26,32)(21,29,24,30)

Polycyclic presentation:

2^9|b^2=i,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7836655);