#8745825

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4515 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)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,26,31,33)(27,34,29,32)(28,30,35,40)(36,37,38,39);(2,3)(6,7)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,27,31,29)(26,32,33,34)(28,38,35,36)(30,37,40,39);(1,2,5,3)(4,6,8,7)(12,20)(17,22)(19,21)(23,24)(26,33)(28,35)(32,34)(36,38);(1,3,5,2)(4,7,8,6)(9,12,15,20)(10,17,14,22)(11,19,13,21)(16,23,18,24)(25,28)(26,30)(27,36)(29,38)(31,35)(32,39)(33,40)(34,37);(1,4,5,8)(2,6,3,7)(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,29,31,27)(26,34,33,32)(28,38,35,36)(30,37,40,39);(9,14,15,10)(11,18,13,16)(12,17,20,22)(19,23,21,24)(25,30)(26,35)(27,37)(28,33)(29,39)(31,40)(32,36)(34,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653021);