#476704

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,10,17,12,18,11,16)(7,19,22,13,23,39,36,40)(8,24,27,44,29,43,28,14)(9,30,33,47,35,46,34,15)(20,50,25,41,52,37,56,48)(21,53,55,49,31,38,63,62)(26,58,32,45,61,42,60,57)(51,67,54,66,68,65,59,64);(1,2)(3,4)(5,7,10,22,12,23,11,36)(6,13,16,19,18,40,17,39)(8,25,27,52,29,56,28,20)(9,31,33,63,35,21,34,55)(14,41,43,50,44,48,24,37)(15,38,46,49,47,53,30,62)(26,59,32,51,61,54,60,68)(42,65,45,66,58,67,57,64);(2,4)(5,8,12,29)(6,14,18,44)(7,20,23,52)(9,32,35,60)(10,28,11,27)(13,37,40,50)(15,45,47,57)(16,24,17,43)(19,48,39,41)(21,54,31,59)(22,56,36,25)(26,34,61,33)(30,58,46,42)(38,65,53,67)(49,64,62,66)(51,63,68,55);(5,9,11,34,12,35,10,33)(6,15,17,30,18,47,16,46)(7,21,36,63,23,31,22,55)(8,26,28,60,29,61,27,32)(13,38,39,62,40,53,19,49)(14,42,24,57,44,58,43,45)(20,51,56,59,52,68,25,54)(37,64,48,67,50,66,41,65)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383900);