#6357751

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,8,11)(3,5,10,12)(4,7,15,6)(9,13,16,14)(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);(1,3,6,13,8,10,7,14)(2,9,15,12,11,16,4,5)(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);(1,4,7,11,8,15,6,2)(3,12,14,9,10,5,13,16)(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);(1,5)(2,10)(3,11)(4,13)(6,9)(7,16)(8,12)(14,15)(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);(1,6,8,7)(2,4,11,15)(3,13,10,14)(5,16,12,9)(20,33)(24,40)(29,30)(34,45)(37,38)(41,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=i,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, 6264947);