#8509904

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

4376 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8417100);