#8181134

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

4983 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8088330);