#498165

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂×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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405361);