#10143473

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

4839 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=ghi,e^2=h,f^2=gh,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, 10050669);