#508955

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

581 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,27,40,36,44,38,47,24,28,25,30,26,31,42,52)(23,32,49,45,39,51,37,29,33,50,34,48,35,46,41,43);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(22,28)(29,43)(30,40)(31,44)(32,50)(45,48)(46,51)(47,52);(2,4)(8,16)(11,18)(14,20)(17,19)(21,23)(22,29)(24,33)(25,37)(26,39)(27,41)(28,43)(30,46)(31,48)(32,47)(34,38)(35,36)(40,51)(42,49)(44,45)(50,52);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(23,33)(30,40)(31,44)(34,49)(35,39)(37,41)(47,52)

Polycyclic presentation:

2^9|a^2=ef,b^2=egi,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416151);