#8936698

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

5700 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)(11,12)(13,14)(15,16)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31);(1,2)(3,6)(4,5)(7,8)(10,14)(12,15)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32);(2,5)(3,7)(9,11,13,16)(10,15,14,12)(17,19,23,20)(18,25,22,24)(21,28,30,27)(26,31,29,32);(3,7)(6,8)(10,14)(12,15)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,15)(10,11,14,16)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8843894);