#8993772

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

4514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,19,22)(12,17,21,24)(13,20,23,18)(26,31)(27,33)(32,38)(34,39);(2,3)(6,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,21)(23,24)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,38)(39,40);(1,3,5,2)(4,7,8,6)(10,16)(11,19)(12,21)(13,23)(27,33)(28,35)(29,37)(32,38);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,21)(23,24)(25,27,30,33)(26,28,31,35)(29,34,37,39)(32,36,38,40);(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,24)(18,19)(25,28)(26,27)(29,36)(30,35)(31,33)(32,34)(37,40)(38,39);(9,13,14,23)(10,18,16,20)(11,17,19,24)(12,22,21,15)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=ghi,f^2=h,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, 8900968);