#494519

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1282 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fhi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401715);