#494313

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1046 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)(8,26)(9,16)(10,18)(11,17)(12,34)(13,19)(14,23)(15,29)(21,52)(22,50)(24,38)(25,51)(27,47)(28,42)(30,62)(31,46)(32,48)(33,44)(35,39)(36,43)(37,56)(40,60)(41,58)(45,63)(49,55)(53,66)(54,61)(57,67)(59,65)(64,68);(1,3,4,2)(5,7)(6,14)(8,21)(9,22)(10,24)(11,23)(12,35)(13,25)(15,37)(16,38)(17,39)(18,20)(19,41)(26,61)(27,53)(28,64)(29,59)(30,57)(31,58)(32,60)(33,55)(34,40)(36,56)(42,52)(43,68)(44,67)(45,66)(46,50)(47,49)(48,51)(54,63)(62,65);(5,8)(6,15)(7,21)(9,27)(10,29)(11,33)(12,36)(13,30)(14,37)(16,42)(17,26)(18,47)(19,45)(20,49)(22,53)(23,55)(24,59)(25,57)(28,31)(32,63)(34,44)(35,56)(38,52)(39,61)(40,67)(41,66)(43,48)(46,62)(50,65)(51,68)(54,60)(58,64);(5,9,13,12)(6,16,19,34)(7,22,25,35)(8,27,30,36)(10,31,32,11)(14,38,41,40)(15,42,45,44)(17,18,46,48)(20,50,51,39)(21,53,57,56)(23,24,58,60)(26,47,62,43)(28,63,33,29)(37,52,66,67)(49,65,68,61)(54,55,59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401509);