#6357743

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62527	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1847 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:

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

Polycyclic presentation:

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

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, 6264939);