#8108708

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

4914 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8015904);