#479841

order	512
nilpotency class	3
isoclinism	isoclinic class of #479840
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17273
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

757 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,8,15,31)(4,17)(5,23,26,10)(6,27)(7,11)(9,21)(12,19)(13,38,18,47)(14,42)(16,46,39,30)(20,48,55,63)(22,49)(24,56)(25,37,52,57)(28,50,32,41)(29,44)(33,60,36,54)(34,51)(35,45)(40,62)(43,61)(53,58)(59,64);(1,3)(2,8,11,31)(4,18,22,13)(5,24)(6,16)(7,15)(9,32)(10,35,23,56)(12,30,27,46)(14,25)(17,47)(19,39)(20,53,55,64)(21,50,51,41)(26,45)(28,34)(29,37,42,57)(33,61)(36,62)(38,49)(40,54,43,60)(44,52)(48,58)(59,63);(1,4,19,51,7,22,6,21)(2,9,27,49,11,34,12,17)(3,13,39,50,15,18,16,41)(5,20,52,60,26,55,25,54)(8,28,46,47,31,32,30,38)(10,36,57,48,23,33,37,63)(14,43,24,53,44,40,45,64)(29,58,56,61,42,59,35,62);(1,5,6,25,7,26,19,52)(2,10,27,57,11,23,12,37)(3,14,39,24,15,44,16,45)(4,20,21,54,22,55,51,60)(8,29,30,35,31,42,46,56)(9,33,49,63,34,36,17,48)(13,40,50,64,18,43,41,53)(28,58,38,62,32,59,47,61)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387037);