#8969615

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

4646 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=gh,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, 8876811);