#8253921

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

4629 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,e^2=i,f^2=gi,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, 8161117);