#503122

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
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	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1251 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1251 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,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, 410318);