#503562

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 731 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,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

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, 410758);