#263503

order	512
nilpotency class	3
isoclinism	isoclinic class of #263499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

838 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,5)(4,7)(6,8)(9,10,16,22)(11,23,27,47)(12,18,31,44)(13,32,35,45)(14,21,15,20)(17,36,40,26)(19,34,46,33)(24,41,53,49)(25,54,56,39)(28,38,52,64)(29,58,60,68)(30,48,37,43)(42,62,66,59)(50,61,67,55)(51,63,71,72)(57,69,70,65);(3,8)(5,6)(9,11,15,36,16,27,14,26)(10,17,21,47,22,40,20,23)(12,28,37,53,31,52,30,24)(13,25,34,61,35,56,33,55)(18,41,48,64,44,49,43,38)(19,39,45,67,46,54,32,50)(29,57,62,71,60,70,59,51)(42,65,68,72,66,69,58,63);(2,5)(3,8)(6,7)(9,12)(10,18)(11,24)(13,29)(14,30)(15,37)(16,31)(17,38)(19,42)(20,43)(21,48)(22,44)(23,49)(25,51)(26,52)(27,53)(28,36)(32,58)(33,59)(34,62)(35,60)(39,63)(40,64)(41,47)(45,68)(46,66)(50,69)(54,72)(55,70)(56,71)(57,61)(65,67);(9,13,15,34,16,35,14,33)(10,19,21,45,22,46,20,32)(11,25,36,61,27,56,26,55)(12,29,37,62,31,60,30,59)(17,39,47,67,40,54,23,50)(18,42,48,68,44,66,43,58)(24,51,28,57,53,71,52,70)(38,63,41,65,64,72,49,69)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170699);