#8291694

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

4972 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)(3,4)(10,13)(14,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32);(1,2,6,5)(3,8,4,7)(11,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(19,25)(23,26)(27,31)(30,32);(1,4,6,3)(2,8,5,7)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32);(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198890);