#8634059

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	#37331	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4491 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)(9,10,13,14)(11,15,19,17)(12,16,21,22)(18,23,20,24)(26,31)(28,32)(33,39)(36,37);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,17,19,15)(12,16,21,22)(18,24,20,23)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(1,3)(2,5)(4,7)(6,8)(10,14)(15,17)(16,22)(23,24)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,21)(22,23)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,39)(38,40);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8541255);