#6607843

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10,13,15)(11,16,12,14)(17,18,22,26)(19,23,28,36)(20,30,31,45)(21,33,34,25)(24,38,39,32)(27,40,41,48)(29,44,43,37)(35,46,47,42);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,28)(18,23,26,36)(20,27,31,41)(21,29,34,43)(24,35,39,47)(25,37,33,44)(30,40,45,48)(32,42,38,46);(11,12)(14,16)(17,20,21,32,22,31,34,38)(18,24,25,30,26,39,33,45)(19,27,29,42,28,41,43,46)(23,35,37,40,36,47,44,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(19,28)(23,36)(27,41)(29,43)(35,47)(37,44)(40,48)(42,46);(11,12)(14,16)(19,28)(20,31)(23,36)(24,39)(29,43)(30,45)(32,38)(37,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6515039);