#503168

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1118 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)(5,6,9,15)(7,19,21,13)(8,25,27,59)(10,17,31,47)(11,16,32,46)(12,35,33,18)(14,41,43,28)(20,52,55,67)(22,39,34,51)(23,38,54,48)(24,40,36,49)(26,61,62,58)(29,60,53,45)(30,64,63,44)(37,56,50,65)(42,57,68,66);(1,2,4,3)(5,7,9,21)(6,13,15,19)(8,26,27,62)(10,23,31,54)(11,34,32,22)(12,36,33,24)(14,42,43,68)(16,39,46,51)(17,48,47,38)(18,49,35,40)(20,53,55,29)(25,58,59,61)(28,57,41,66)(30,65,63,56)(37,64,50,44)(45,52,60,67);(5,8,11,29,12,30,31,41,9,27,32,53,33,63,10,28)(6,14,17,44,18,45,46,59,15,43,47,64,35,60,16,25)(7,20,54,62,24,57,34,65,21,55,23,26,36,66,22,56)(13,37,51,68,40,58,48,52,19,50,39,42,49,61,38,67);(1,3,4,2)(6,15)(8,27)(13,19)(16,46)(17,47)(18,35)(20,55)(26,62)(28,41)(29,53)(30,63)(38,48)(39,51)(40,49)(56,65)(57,66)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,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,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410364);