#9003075

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

4482 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^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, 8910271);