#403333

order	512
nilpotency class	3
isoclinism	isoclinic class of #403332
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

2246 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 2246 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)(3,13)(4,18)(5,10)(6,24)(7,12)(8,16)(9,21)(11,26)(14,36)(15,42)(17,38)(19,47)(20,32)(22,35)(23,34)(25,33)(27,40)(28,43)(29,46)(30,55)(31,50)(37,44)(39,61)(41,57)(45,54)(48,52)(49,58)(51,59)(53,62)(56,64)(60,63);(1,3)(2,8,12,29)(4,14)(5,15)(6,16)(7,17)(9,30,18,27)(10,28,34,54)(11,13,24,38)(19,39)(20,40)(21,41)(22,43)(23,44)(25,45)(26,46)(31,56,47,53)(32,57,48,36)(33,42,35,37)(49,62)(50,63)(51,64)(52,55)(58,60,59,61);(1,4)(2,9,12,18)(3,14)(5,19)(6,20)(7,21)(8,27,29,30)(10,31,34,47)(11,32,24,48)(13,36,38,57)(15,39)(16,40)(17,41)(22,49)(23,50)(25,51)(26,52)(28,53,54,56)(33,58,35,59)(37,60,42,61)(43,62)(44,63)(45,64)(46,55);(1,5,7,23)(2,10,12,34)(3,15,17,44)(4,19,21,50)(6,25,26,22)(8,28,29,54)(9,31,18,47)(11,35,24,33)(13,37,38,42)(14,39,41,63)(16,45,46,43)(20,51,52,49)(27,53,30,56)(32,59,48,58)(36,60,57,61)(40,64,55,62)

Polycyclic presentation:

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

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, 310529);