#6355056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355044
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	C₄²	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1581 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=eh,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, 6262252);