#8919248

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

4867 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^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, 8826444);