#6603320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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)(3,7)(4,8)(5,6)(17,18,22,25)(19,23,28,37)(20,30,32,40)(21,33,35,26)(24,34,39,31)(27,41,42,48)(29,45,44,38)(36,46,47,43);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,28)(18,23,25,37)(20,27,32,42)(21,29,35,44)(24,36,39,47)(26,38,33,45)(30,41,40,48)(31,43,34,46);(10,13)(11,15)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(26,40,33,30)(29,46,44,43)(38,48,45,41);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(18,25)(23,37)(24,39)(26,33)(30,40)(36,47)(38,45)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6510516);