#8818977

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

4695 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8726173);