#8436572

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	#36808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 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)(26,31)(27,36)(34,37)(35,40);(1,3)(2,7)(4,8)(5,6)(9,11)(10,16)(12,14)(13,21)(15,22)(17,18)(19,20)(23,24)(25,26,30,31)(27,32,36,33)(28,34,38,37)(29,35,39,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,13)(10,17,19,23)(11,14,22,21)(16,18,20,24)(25,27)(26,32)(28,35)(29,37)(30,36)(31,33)(34,39)(38,40);(1,5,6,3)(2,4,8,7)(9,13,15,12)(10,17,19,23)(11,21,22,14)(16,18,20,24)(25,27)(26,33)(28,35)(29,34)(30,36)(31,32)(37,39)(38,40);(9,14,15,21)(10,18,19,24)(11,12,22,13)(16,17,20,23)(25,28,30,38)(26,34,31,37)(27,35,36,40)(29,32,39,33);(1,4)(2,5)(3,8)(6,7)(9,10)(11,16)(12,17)(13,23)(14,18)(15,19)(20,22)(21,24)(25,29)(26,35)(27,37)(28,32)(30,39)(31,40)(33,38)(34,36)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=gh,e^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, 8343768);