#508280

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

733 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 76
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 76 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,23)(18,24)(19,22)(25,40)(26,42)(27,43)(28,41)(29,44)(30,46)(31,47)(32,45)(33,50)(34,51)(35,49)(36,52)(37,48)(38,53)(39,54)(55,66)(56,68)(57,67)(58,70)(59,71)(60,69)(61,72)(62,73)(63,74)(64,75)(65,76);(6,9)(10,12)(13,15,19,28)(14,20,22,41)(16,29,32,63)(17,33,37,62)(18,38,35,27)(21,44,45,74)(23,50,48,73)(24,53,49,43)(25,55,57,30)(26,58,60,36)(31,64,39,59)(34,61,65,56)(40,66,67,46)(42,70,69,52)(47,75,54,71)(51,72,76,68);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,19,32)(14,21,22,45)(15,25,28,57)(17,34,37,65)(18,39,35,31)(20,40,41,67)(23,51,48,76)(24,54,49,47)(26,59,60,64)(27,61,38,56)(29,62,63,33)(30,58,55,36)(42,71,69,75)(43,72,53,68)(44,73,74,50)(46,70,66,52);(7,11)(10,12)(14,22)(20,41)(21,45)(23,48)(24,49)(40,67)(42,69)(43,53)(44,74)(46,66)(47,54)(50,73)(51,76)(52,70)(68,72)(71,75)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=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, 415476);