#8230863

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

4894 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)(17,18,22,23)(19,25,26,21)(20,28,29,24)(27,30,32,31);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,26)(20,29)(30,31);(2,6)(4,7)(9,10)(11,13)(12,14)(15,16)(18,23)(21,25)(24,28)(30,31);(9,11)(10,13)(12,15)(14,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,29)(18,24,23,28)(19,27,26,32)(21,30,25,31);(1,4,5,7)(2,3,6,8)(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,e^2=g,f^2=i,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, 8138059);