#7736170

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

4881 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7643366);