#8108029

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8015225);