#8458477

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

4986 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)(3,8)(10,13)(11,15)(17,18)(19,24)(20,31)(21,26)(22,27)(23,28)(25,30)(29,32);(1,2,5,6)(3,4,8,7)(9,10,12,13)(11,14,15,16)(17,19,23,27)(18,24,28,22)(20,21,30,29)(25,32,31,26);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,30)(18,25,28,31)(19,29,27,21)(22,32,24,26);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,26)(31,32);(9,11,12,15)(10,14,13,16)(17,21,23,29)(18,26,28,32)(19,30,27,20)(22,31,24,25);(1,4,5,7)(2,8,6,3)(17,22,23,24)(18,27,28,19)(20,32,30,26)(21,31,29,25)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365673);