#9123793

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

4753 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9030989);