#511133

order	512
nilpotency class	4
isoclinism	isoclinic class of #511133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32260
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 118 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)(5,26)(6,13)(7,12)(8,14)(9,17)(10,23)(11,19)(16,33)(18,46)(21,36)(22,40)(24,43)(25,38)(27,35)(28,57)(29,34)(30,59)(31,42)(32,44)(37,47)(39,55)(41,51)(45,48)(49,61)(50,62)(52,53)(54,64)(56,60)(58,63);(1,3,5,17,8,19,7,27)(2,9,12,15,14,35,26,11)(4,21,23,50,25,54,22,53)(6,29,28,51,32,58,31,18)(10,36,20,52,40,64,38,62)(13,41,42,34,44,46,57,63)(16,30,48,24,49,55,47,56)(33,60,37,39,61,43,45,59);(1,4,5,22,8,25,7,23)(2,10,26,20,14,40,12,38)(3,16,27,48,19,49,17,47)(6,30,28,56,32,55,31,24)(9,33,15,45,35,61,11,37)(13,43,57,59,44,60,42,39)(18,50,58,54,51,53,29,21)(34,62,46,64,63,52,41,36);(1,3)(2,11)(4,16)(5,27)(6,18)(7,17)(8,19)(9,26)(10,37)(12,35)(13,41)(14,15)(20,33)(21,24)(22,48)(23,47)(25,49)(28,58)(29,31)(30,50)(32,51)(34,57)(36,43)(38,61)(39,52)(40,45)(42,63)(44,46)(53,55)(54,56)(59,62)(60,64)

Polycyclic presentation:

2^9|b^2=f,c^2=efh,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi

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, 418329);