#6355077

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	#60948	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1599 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(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)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(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);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(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);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(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|b^2=f,d^2=hi,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, 6262273);