#8213915

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

4919 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8121111);