#9002409

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

4375 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)(9,10,14,17)(11,15,18,13)(12,16,20,23)(19,21,24,22)(25,26)(27,35)(28,33)(29,39)(30,31)(32,37)(34,40)(36,38);(3,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,21)(20,24)(22,23)(26,31)(28,38)(32,35)(34,39);(1,3,5,8)(2,6,4,7)(11,18)(12,20)(13,15)(16,23)(25,27,30,37)(26,32,31,35)(28,39,38,34)(29,33,40,36);(3,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,20)(15,21)(17,23)(18,24)(25,27)(26,32)(28,39)(29,36)(30,37)(31,35)(33,40)(34,38);(1,3)(2,7)(4,6)(5,8)(9,13,14,15)(10,11,17,18)(12,21,20,22)(16,24,23,19)(25,28,30,38)(26,33,31,36)(27,34,37,39)(29,35,40,32);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,21)(20,24)(22,23)(25,29)(26,34)(27,36)(28,35)(30,40)(31,39)(32,38)(33,37)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,e^2=hi,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, 8909605);