#8634052

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

4523 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8541248);