#503565

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=efh,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410761);