#8919504

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

5094 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

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, 8826700);