#8756100

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

4374 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)(3,7)(4,8)(5,6)(9,10,13,14)(11,17,18,15)(12,16,20,21)(19,23,24,22)(27,33)(31,34)(35,39)(38,40);(1,2,5,6)(3,7,8,4)(10,14)(12,20)(15,17)(19,24)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,3)(2,4)(5,8)(6,7)(11,18)(15,17)(19,24)(22,23)(26,30)(31,34)(32,36)(38,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663296);