#6355100

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

1518 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=hi,e^2=i,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, 6262296);