#503522

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

775 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 174 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)(3,7,4,6)(9,10,16,22)(11,23,27,51)(12,28,34,45)(13,35,15,39)(14,38,36,20)(17,41,44,26)(18,31,47,33)(19,30,21,37)(24,52,29,60)(25,48,40,43)(32,66,53,58)(42,68,70,72)(46,64,59,65)(49,62,71,55)(50,61,67,54)(56,69,63,57);(1,3,5,4)(2,6,8,7)(9,11,14,26,16,27,36,41)(10,17,20,23,22,44,38,51)(12,29,32,63,34,24,53,56)(13,25,37,54,15,40,30,61)(18,42,46,49,47,70,59,71)(19,43,35,50,21,48,39,67)(28,57,58,60,45,69,66,52)(31,62,65,68,33,55,64,72);(1,3,5,4)(2,7,8,6)(9,12,30,64,14,32,13,31,16,34,37,65,36,53,15,33)(10,18,39,58,20,46,19,45,22,47,35,66,38,59,21,28)(11,24,54,72,26,56,40,62,27,29,61,68,41,63,25,55)(17,42,50,52,23,49,48,57,44,70,67,60,51,71,43,69);(1,4,5,3)(2,7,8,6)(9,13,36,30,16,15,14,37)(10,19,38,39,22,21,20,35)(11,25,41,61,27,40,26,54)(12,31,53,64,34,33,32,65)(17,43,51,67,44,48,23,50)(18,45,59,58,47,28,46,66)(24,55,63,68,29,62,56,72)(42,69,71,60,70,57,49,52)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,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, 410718);