#8230878

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

4830 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,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, 8138074);