#500729

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
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	#61090	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1040 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407925);