#8741686

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4258 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8648882);