#8898591

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

4592 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)(6,8)(9,10)(11,18)(12,17)(13,14)(15,16)(19,24)(20,21)(22,23)(25,26,30,33)(27,34,29,31)(28,32,36,40)(35,39,37,38);(3,7)(6,8)(10,16)(13,19)(17,18)(20,22)(27,29)(31,34)(35,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,18)(13,20,19,22)(14,23,24,21)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,40,39);(9,12,15,11)(10,18,16,17)(13,20,19,22)(14,23,24,21)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,13,15,19)(10,14,16,24)(11,20,12,22)(17,23,18,21)(25,29,30,27)(26,31,33,34)(28,35,36,37)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,21)(12,23)(13,16)(15,24)(17,20)(18,22)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805787);