#410822

order	512
nilpotency class	3
isoclinism	isoclinic class of #410821
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15091
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

1195 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1195 conjugacy classes of subgroups — too many to draw. Only its 232 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318018);