#8436701

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	#53423	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4413 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=gh,f^2=g,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, 8343897);