#8528077

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

4412 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)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(3,7)(5,8)(11,12)(15,18)(19,20)(23,24)(26,28)(27,34)(32,36)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,17,18)(13,20,22,19)(16,24,21,23)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3,4,7)(2,5,6,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39);(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=h,e^2=g,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, 8435273);