#8634411

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

4465 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541607);