#8228634

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

4355 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:

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

Polycyclic presentation:

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

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, 8135830);