#502199

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
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	#33527	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409395);