#8926512

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	#47004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26)(27,33)(28,30)(29,36)(31,35)(32,37)(34,38)(39,40);(3,7)(5,8)(9,11)(10,16)(12,20)(13,17)(14,19)(15,21)(18,23)(22,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2)(3,5)(4,6)(7,8)(9,12,15,23)(10,17,19,24)(11,20,21,18)(13,14,22,16)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,13)(10,18)(11,17)(12,14)(15,22)(16,23)(19,20)(21,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(1,3)(2,5)(4,7)(6,8)(9,14)(10,11)(12,22)(13,23)(15,16)(17,20)(18,24)(19,21)(25,27)(26,33)(28,31)(29,34)(30,35)(32,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,19)(20,23)(22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833708);