#8188148

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4405 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,20)(14,16)(15,19)(17,22)(18,23)(21,24)(26,30)(27,33)(32,37)(34,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2,4,6)(3,5,8,7)(10,16)(13,15)(17,20)(21,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,12,14,22)(10,17,16,20)(11,18,19,24)(13,21,15,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,15)(10,11,16,19)(12,21,22,23)(17,18,20,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8095344);