#8181161

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

4582 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=g,e^2=g,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, 8088357);