#8434190

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	#38510	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4635 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341386);