#6348807

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62816	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2792 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256003);