#8196403

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

4876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103599);