#501367

order	512
nilpotency class	4
isoclinism	isoclinic class of #501366
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24531
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33522	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

657 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,13,18,8)(4,19,21,38)(5,11)(6,27)(7,10)(9,22,39,24)(12,30)(14,45)(15,34,17,32)(16,33,50,44)(20,52)(23,56,55,42)(25,58)(26,36)(28,41,48,60)(29,37,57,64)(31,47)(35,49)(40,59)(43,51)(46,61)(53,62)(54,63);(1,3)(2,8)(4,20)(5,15)(6,28)(7,18)(9,35)(10,13)(11,32)(12,41)(14,23)(16,25)(17,26)(19,52)(21,53)(22,54)(24,49)(27,60)(29,47)(30,48)(31,37)(33,40)(34,36)(38,62)(39,63)(42,61)(43,64)(44,58)(45,56)(46,55)(50,59)(51,57);(1,4,5,22,7,21,26,24)(2,9,36,38,10,39,11,19)(3,14,17,51,18,46,15,47)(6,29,25,55,30,57,59,23)(8,31,32,61,13,43,34,45)(12,42,58,37,27,56,40,64)(16,49,28,53,50,54,48,20)(33,62,60,35,44,52,41,63);(1,3)(2,8)(4,14)(5,17)(6,16)(7,18)(9,31)(10,13)(11,34)(12,33)(15,26)(19,45)(20,23)(21,46)(22,51)(24,47)(25,28)(27,44)(29,49)(30,50)(32,36)(35,37)(38,61)(39,43)(40,41)(42,62)(48,59)(52,56)(53,55)(54,57)(58,60)(63,64)

Polycyclic presentation:

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

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, 408563);