#10106993

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

4944 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gh,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, 10014189);