#8942492

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

4419 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)(25,26,31,33)(27,34,29,28)(30,32,39,40)(35,38,37,36);(1,3,5,8)(2,6,7,4)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27,31,29)(26,28,33,34)(30,35,39,37)(32,36,40,38);(1,2)(3,6)(4,8)(5,7)(10,15)(12,19)(13,21)(18,23)(25,28,31,34)(26,27,33,29)(30,38,39,36)(32,37,40,35);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,26)(27,28)(29,34)(30,32)(31,33)(35,36)(37,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,27)(26,28,33,34)(30,37,39,35)(32,36,40,38);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,39)(26,32,33,40)(27,35,29,37)(28,36,34,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8849688);