#503154

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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	#61086	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

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, 410350);