#10156490

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

5138 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)(3,7)(4,5)(6,8)(18,19)(23,25)(24,26)(31,32);(1,2,5,4)(3,6,8,7)(9,10)(11,16)(12,14)(13,15)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,14,15,16);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,16)(13,15)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,14,15,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)

Polycyclic presentation:

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

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, 10063686);