#6355604

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1499 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18,23,28)(19,29,31,40)(20,34,35,42)(21,27,22,26)(24,32,39,33)(25,37,41,36)(30,44,47,38)(43,45,48,46);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19,21,33,23,31,22,32)(18,24,26,29,28,39,27,40)(20,30,36,46,35,47,37,45)(25,38,34,43,41,44,42,48);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,16)(12,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,34)(27,42)(28,41)(29,43)(31,47)(32,45)(33,46)(39,44)(40,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(19,31)(20,35)(24,39)(25,41)(29,40)(32,33)(34,42)(36,37);(3,12)(7,15)(10,16)(11,13)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,35,37)(24,40,39,29)(25,34,41,42)(30,45,47,46)(38,48,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262800);