#8093318

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	#45514	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4332 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=g,e^2=hi,f^2=g,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, 8000514);