#9018803

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

4869 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)(3,8)(4,7)(5,6)(10,13)(14,16)(17,18,22,24)(19,27,28,23)(20,29,30,25)(21,26,31,32);(3,4)(7,8)(11,15)(14,16)(17,19,22,28)(18,23,24,27)(20,21,30,31)(25,26,29,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(18,24)(20,30)(21,31)(23,27);(3,4)(7,8)(17,20,22,30)(18,25,24,29)(19,21,28,31)(23,26,27,32);(1,4,6,3)(2,7,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,20)(22,31)(23,25)(24,32)(27,29)(28,30);(1,5,6,2)(3,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=gh,f^2=i,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, 8925999);