#8701866

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

5000 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8609062);