#8994054

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

4753 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)(10,13)(11,15)(17,18)(19,24)(20,21)(22,26)(23,27)(25,32)(28,31)(29,30);(1,2,4,6)(3,5,7,8)(9,10,14,13)(11,16,15,12)(19,27)(23,24)(28,30)(29,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19,22,27)(18,23,26,24)(20,30,25,28)(21,31,32,29);(9,10,14,13)(11,12,15,16)(17,20,22,25)(18,21,26,32)(19,28,27,30)(23,29,24,31);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,15,13,11)(17,19)(18,24)(20,30)(21,29)(22,27)(23,26)(25,28)(31,32);(1,3)(2,5)(4,7)(6,8)(9,13,14,10)(11,16,15,12)(17,21)(18,25)(19,29)(20,26)(22,32)(23,30)(24,28)(27,31)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8901250);