#510029

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=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, 417225);