#8115324

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	#37845	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4337 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022520);