#9040985

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
prime radical	#51902	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4966 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)(11,14)(12,13)(15,16)(17,18,23,22)(19,27,29,24)(20,26,21,25)(28,31,30,32);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,28)(21,30)(22,27)(23,29)(25,31)(26,32);(2,5)(3,7)(10,12)(11,15)(17,20,23,21)(18,25,22,26)(19,28,29,30)(24,31,27,32);(2,5)(6,8)(11,15)(14,16)(17,21,23,20)(18,26,22,25)(19,28,29,30)(24,31,27,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,22,23,18)(19,27,29,24)(20,25,21,26)(28,31,30,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,19)(18,24)(20,30)(21,28)(22,27)(23,29)(25,32)(26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948181);