#8362458

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	#40504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8269654);