#8993615

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

4481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=ghi,f^2=gh,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, 8900811);