#8463185

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=i,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, 8370381);