#8689284

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

4998 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,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, 8596480);