#10158343

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

4918 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,e^2=hi,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, 10065539);