#512399

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,h^a=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419595);