#6481481

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

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

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, 6388677);