#317184

order	512
nilpotency class	3
isoclinism	isoclinic class of #317171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

560 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,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=gh,c^b=cg,f^b=fh,d^c=di,f^c=fi,g^c=gh

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, 224380);