#8093138

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	#45726	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4574 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^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, 8000334);