#403339

order	512
nilpotency class	3
isoclinism	isoclinic class of #403338
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14873
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2240 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 2240 conjugacy classes of subgroups — too many to draw. Only its 280 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,e^d=ei,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310535);