#8651470

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=hi,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, 8558666);