#8550919

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

4498 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458115);