#8692646

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

5555 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,25,26,21)(20,24,29,31)(27,28,32,30);(2,3)(6,7)(11,15)(13,16)(18,23)(19,26)(24,31)(27,32);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,19,22,26)(18,21,23,25)(20,27,29,32)(24,30,31,28);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(17,21)(18,19)(20,28)(22,25)(23,26)(24,32)(27,31)(29,30)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8599842);