#8758066

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	#52004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4756 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8665262);