#8527399

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	#40606	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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)(11,21)(12,18)(13,15)(14,19)(16,20)(17,24)(22,23)(25,26,31,30)(27,32,37,36)(28,38,33,29)(34,40,39,35);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,22)(13,19)(14,18)(15,23)(16,24)(20,21)(25,27,31,37)(26,32,30,36)(28,34,33,39)(29,35,38,40);(1,2,4,6)(3,5,7,8)(9,12,16,13)(10,18,20,15)(11,22,24,19)(14,21,23,17)(25,28)(26,29)(27,34)(30,38)(31,33)(32,35)(36,40)(37,39);(3,7)(5,8)(9,13,16,12)(10,18,20,15)(11,19,24,22)(14,21,23,17)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,14,16,23)(10,19,20,22)(11,18,24,15)(12,17,13,21)(25,29,31,38)(26,33,30,28)(27,35,37,40)(32,39,36,34);(1,3)(2,5)(4,7)(6,8)(9,15,16,18)(10,13,20,12)(11,23,24,14)(17,19,21,22)(25,30,31,26)(27,36,37,32)(28,38,33,29)(34,40,39,35)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8434595);