#8992088

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	#41210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4233 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8899284);