#8341817

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

4922 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)(19,26,21,24)(20,28,30,25)(27,31,29,32);(1,2,5,4)(3,6,7,8)(10,14)(11,15)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31);(2,4)(6,8)(9,11,13,15)(10,12,14,16)(18,22)(24,26)(25,28)(31,32);(3,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32);(1,3,5,7)(2,6,4,8)(17,21,23,19)(18,26,22,24)(20,27,30,29)(25,31,28,32);(1,4,5,2)(3,8,7,6)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8249013);