#8968103

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

4689 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=hi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8875299);