#8688260

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

4556 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,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, 8595456);