#6484601

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473158
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6391797);