#8968195

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

5872 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)(3,7)(4,6)(5,8)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32);(1,2,4,6)(3,5,8,7)(9,10,13,12)(11,15,16,14)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30);(1,2,4,6)(3,7,8,5)(9,11,13,16)(10,14,12,15)(18,22)(19,21)(25,28)(27,30);(1,2,4,6)(3,5,8,7)(9,11,13,16)(10,15,12,14)(17,20,23,29)(18,25,22,28)(19,27,21,30)(24,31,26,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,21,23,19)(18,26,22,24)(20,27,29,30)(25,31,28,32);(1,3,4,8)(2,5,6,7)(9,12,13,10)(11,14,16,15)(17,22,23,18)(19,26,21,24)(20,28,29,25)(27,32,30,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=gi,e^2=gh,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, 8875391);