#8186888

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

4489 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094084);