#8970647

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

4382 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8877843);