#8688244

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

4389 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8595440);