#503054

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

721 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,5)(3,7,8,6)(9,10,16,21)(11,17,25,42)(12,26,29,57)(13,32,34,62)(14,20,15,37)(18,43,45,30)(19,33,48,28)(22,49,51,71)(23,54,55,67)(24,40,35,41)(27,56,53,72)(31,58,36,59)(38,65,66,52)(39,63,68,50)(44,70,69,60)(46,64,47,61);(2,5)(6,7)(9,11)(10,17)(12,27)(13,23)(14,35)(15,24)(16,25)(18,44)(19,39)(20,41)(21,42)(22,31)(26,56)(28,50)(29,53)(30,60)(32,54)(33,63)(34,55)(36,51)(37,40)(38,46)(43,70)(45,69)(47,66)(48,68)(49,58)(52,61)(57,72)(59,71)(62,67)(64,65);(9,12,28,61,15,31,34,43,16,29,33,64,14,36,13,30)(10,18,32,59,20,47,48,57,21,45,62,58,37,46,19,26)(11,22,50,70,24,53,55,65,25,51,63,60,35,27,23,52)(17,38,54,56,41,44,68,71,42,66,67,72,40,69,39,49);(1,3,4,8)(2,6,5,7)(12,29)(18,45)(22,51)(26,57)(27,53)(30,43)(31,36)(38,66)(44,69)(46,47)(49,71)(52,65)(56,72)(58,59)(60,70)(61,64)

Polycyclic presentation:

2^9|a^2=ei,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=di,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, 410250);