#8968178

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

5007 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:

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

Polycyclic presentation:

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

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, 8875374);