#6953574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#38025	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26,30,36)(27,37,39,32)(28,40,38,34)(29,35,31,33);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,22)(20,24)(25,27,30,39)(26,32,36,37)(28,31,38,29)(33,40,35,34);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(26,33)(28,38)(29,31)(32,40)(34,37)(35,36);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,28,30,38)(26,34,36,40)(27,29,39,31)(32,35,37,33);(10,14)(15,21)(16,22)(23,24)(25,29,30,31)(26,35,36,33)(27,38,39,28)(32,40,37,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6860770);