#6357699

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
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

1811 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)(3,12)(5,6)(9,13)(10,11)(15,16)(17,18,22,27)(19,28,32,44)(20,25,33,24)(21,35,29,26)(23,36,39,31)(30,43,46,42)(34,40,45,41)(37,47,38,48);(1,2,6,10,7,11,5,8)(3,4,12,16,13,14,9,15)(17,19,29,36,22,32,21,31)(18,23,35,44,27,39,26,28)(20,30,45,48,33,46,34,47)(24,37,41,43,25,38,40,42);(1,3,6,12,7,13,5,9)(2,4,10,16,11,14,8,15)(17,20)(18,24)(19,30)(21,34)(22,33)(23,37)(25,27)(26,40)(28,42)(29,45)(31,47)(32,46)(35,41)(36,48)(38,39)(43,44);(1,4)(2,9)(3,10)(5,15)(6,16)(7,14)(8,13)(11,12)(17,20)(18,25)(19,30)(21,34)(22,33)(23,38)(24,27)(26,41)(28,43)(29,45)(31,47)(32,46)(35,40)(36,48)(37,39)(42,44);(3,13)(4,14)(9,12)(15,16)(20,33)(24,25)(30,46)(34,45)(37,38)(40,41)(42,43)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=ghi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,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, 6264895);