#8979142

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

4687 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)(9,10,15,13)(11,16,14,21)(12,17,19,22)(18,23,20,24)(25,26,30,31)(27,33,35,29)(28,36,37,32)(34,39,40,38);(2,6)(4,7)(9,11,15,14)(10,16,13,21)(12,18,19,20)(17,23,22,24)(25,27,30,35)(26,29,31,33)(28,34,37,40)(32,39,36,38);(1,2)(3,4)(5,6)(7,8)(10,13)(11,14)(12,19)(23,24)(26,31)(28,37)(29,33)(34,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,18)(13,22)(14,20)(15,19)(16,23)(21,24)(25,28)(26,32)(27,34)(29,39)(30,37)(31,36)(33,38)(35,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,10)(11,16,14,21)(12,17,19,22)(18,24,20,23)(25,29)(26,27)(28,38)(30,33)(31,35)(32,40)(34,36)(37,39);(1,3)(2,4)(5,8)(6,7)(9,14,15,11)(10,16,13,21)(12,20,19,18)(17,23,22,24)(25,26)(27,29)(28,32)(30,31)(33,35)(34,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,e^2=gh,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, 8886338);