#8214208

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

5067 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,e^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, 8121404);