#8233362

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	#59568	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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)(11,16)(12,18)(13,20)(14,19)(15,17)(21,24)(22,23)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(3,7)(5,8)(9,11)(10,16)(12,14)(13,17)(15,20)(18,19)(21,22)(23,24)(26,28)(31,33)(32,36)(39,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,20)(14,21)(18,23)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,21)(13,19)(14,20)(15,22)(16,24)(17,23)(25,26,30,28)(27,31,35,33)(29,32,38,36)(34,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,19)(14,23)(15,16)(17,20)(18,21)(22,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(9,14,15,21)(10,19,17,24)(11,12,20,22)(13,23,16,18)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

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

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, 8140558);