#8757460

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

4187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,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, 8664656);