#8455683

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	#53531	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,22)(23,24)(26,32)(27,35)(28,37)(34,40);(2,3)(6,7)(9,11,15,13)(10,16,20,17)(12,23,19,21)(14,22,18,24)(25,26,31,32)(27,33,35,39)(28,38,37,30)(29,34,36,40);(1,2,5,3)(4,6,8,7)(9,11,15,13)(10,17,20,16)(12,21,19,23)(14,22,18,24)(25,27)(26,33)(28,36)(29,37)(30,34)(31,35)(32,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,15,19)(10,18,20,14)(11,21,13,23)(16,22,17,24)(25,28,31,37)(26,30,32,38)(27,36,35,29)(33,34,39,40);(1,4)(2,6)(3,7)(5,8)(9,13,15,11)(10,16,20,17)(12,23,19,21)(14,24,18,22)(25,29,31,36)(26,34,32,40)(27,37,35,28)(30,39,38,33);(9,14,15,18)(10,19,20,12)(11,22,13,24)(16,21,17,23)(25,30)(26,28)(27,34)(29,39)(31,38)(32,37)(33,36)(35,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=h,e^2=hi,f^2=gh,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, 8362879);