#503274

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3415	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1070 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)(5,6,13,19)(7,20,25,59)(8,15,30,51)(9,31,35,48)(10,36,37,42)(11,18,12,40)(14,41,47,23)(16,52,54,22)(17,21,55,32)(24,57,38,58)(26,53,62,65)(27,45,60,46)(28,56,64,67)(29,50,39,44)(33,43,34,66)(49,61,68,63);(5,7,21,60,12,24,10,23,13,25,32,27,11,38,37,41)(6,14,42,57,18,46,17,20,19,47,36,58,40,45,55,59)(8,26,61,34,29,22,28,35,30,62,63,33,39,52,64,9)(15,48,67,16,44,66,49,53,51,31,56,54,50,43,68,65);(1,2,3,4)(5,8)(6,15)(7,22)(9,27)(10,28)(11,39)(12,29)(13,30)(14,43)(16,20)(17,49)(18,44)(19,51)(21,61)(23,33)(24,62)(25,52)(26,38)(31,45)(32,63)(34,41)(35,60)(36,56)(37,64)(40,50)(42,67)(46,48)(47,66)(53,58)(54,59)(55,68)(57,65);(5,9,32,62,12,34,37,52,13,35,21,26,11,33,10,22)(6,16,36,43,18,53,55,48,19,54,42,66,40,65,17,31)(7,8,27,63,24,29,41,64,25,30,60,61,38,39,23,28)(14,44,58,68,46,51,59,67,47,50,57,49,45,15,20,56)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=f,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

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, 410470);