#8581830

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

4562 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8489026);