#9105353

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

4846 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9012549);