#511540

order	512
nilpotency class	4
isoclinism	isoclinic class of #509117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#32652
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2960 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efh,d^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,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, 418736);