#8899006

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

4847 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806202);