#6741047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=gi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648243);