#9064902

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

4789 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972098);