#10075932

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

5209 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)(3,4)(5,6)(7,8)(9,10,13,15)(11,12,16,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,3,6,7)(2,4,5,8)(9,11,13,16)(10,14,15,12)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31);(1,4,6,8)(2,7,5,3)(9,11)(10,12)(13,16)(14,15)(19,25)(22,29)(26,27)(30,31);(1,2,6,5)(3,8,7,4)(9,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,4)(2,3)(5,7)(6,8)(17,19,21,25)(18,22,24,29)(20,27,28,26)(23,31,32,30);(1,5,6,2)(3,8,7,4)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9983128);