#8689966

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

5050 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)(11,13)(12,15)(14,16)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,22,28,27);(1,2)(3,7)(4,6)(5,8)(10,11)(14,15)(17,19)(18,24)(20,27)(21,26)(22,30)(23,31)(25,28)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,11)(12,14,16,15)(17,20)(18,25)(19,22)(21,24)(23,30)(26,29)(27,31)(28,32);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20)(18,26)(19,27)(21,24)(22,31)(23,30)(25,29)(28,32);(1,3)(2,7)(4,8)(5,6)(9,12,13,16)(10,14,11,15)(17,21)(18,27)(19,26)(20,24)(22,29)(23,28)(25,31)(30,32);(1,2)(3,5)(4,6)(7,8)(17,22,23,27)(18,28,29,21)(19,30,31,20)(24,25,32,26)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=g,f^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, 8597162);