#8677259

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	#42616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,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, 8584455);