#8633980

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

4940 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)(3,8)(4,5)(6,7)(10,13)(11,15)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31);(1,3,6,5)(2,4,7,8)(9,10,12,13)(11,14,15,16)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32);(2,7)(4,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32);(2,7)(3,5)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31);(1,5,6,3)(2,4,7,8)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8541176);