#8677417

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=h,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, 8584613);