#9028030

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	#43272	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4389 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935226);