#502509

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efhi,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, 409705);