#501246

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61057	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408442);