#8093257

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(6,8)(12,19)(16,21)(18,23)(22,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(3,7)(6,8)(10,14)(11,17)(16,21)(18,23)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,2,5,4)(3,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24);(1,3)(2,6)(4,8)(5,7)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000453);