#501696

order	512
nilpotency class	4
isoclinism	isoclinic class of #501695
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 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:

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

Polycyclic presentation:

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

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, 408892);