#8810225

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	#36361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717421);