#507943

order	512
nilpotency class	4
isoclinism	isoclinic class of #500592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29533
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 448 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)(4,21)(5,11)(6,28)(7,12)(8,14)(9,18)(10,24)(13,27)(16,45)(17,39)(19,51)(20,34)(22,57)(23,41)(25,55)(26,56)(29,37)(30,42)(31,40)(32,58)(33,49)(35,62)(36,59)(38,60)(43,47)(44,50)(46,54)(48,64)(52,61)(53,63);(1,3)(2,9,14,34)(4,22,26,48)(5,18)(6,19)(7,20)(8,17)(10,35)(11,39,12,15)(13,33,42,51)(16,23,46,25)(21,54)(24,58,60,52)(27,50)(28,44,40,63)(29,47,59,62)(30,53)(31,49)(32,37)(36,61)(38,43)(41,64)(45,56)(55,57);(1,4)(2,10)(3,16)(5,23)(6,29)(7,25)(8,26)(9,32)(11,36)(12,37)(13,41)(14,38)(15,43)(17,46)(18,22)(19,52)(20,48)(21,28)(24,27)(30,60)(31,59)(33,54)(34,61)(35,39)(40,56)(42,55)(44,57)(45,51)(47,50)(49,58)(53,62)(63,64);(2,11,14,12)(3,17)(9,15,34,39)(10,36,38,37)(13,40,42,28)(16,46)(18,20)(19,49)(21,41,56,55)(22,48)(32,43,61,35)(33,44,51,63)(45,64,54,57)(47,62)(50,53)(52,58)

Polycyclic presentation:

2^9|b^2=eg,d^2=gh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,g^c=gi,h^c=hi

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, 415139);