#8125461

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

4712 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8032657);