#8621884

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

4871 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529080);