#8695298

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

4367 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26)(27,32)(28,34)(29,33)(30,35)(31,36)(37,38)(39,40);(2,6)(3,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,31,29)(26,32,36,33)(28,39,35,37)(30,38,34,40);(1,3)(2,4)(5,8)(6,7)(25,27,31,29)(26,33,36,32)(28,37,35,39)(30,38,34,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,10)(11,16,13,18)(14,17,22,20)(19,24,21,23)(25,28,31,35)(26,34,36,30)(27,37,29,39)(32,38,33,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,29,31,27)(26,32,36,33)(28,39,35,37)(30,40,34,38);(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,30,31,34)(26,35,36,28)(27,38,29,40)(32,37,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602494);