#8436766

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	#38420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8343962);