#8679887

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587083);