#9174140

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

4530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9081336);