#410819

order	512
nilpotency class	3
isoclinism	isoclinic class of #410818
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

1203 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1203 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)(3,13,18,33)(4,20,24,9)(5,25,28,10)(6,29,14,11)(8,19,32,17)(15,39,44,34)(16,40,46,57)(21,50,43,30)(22,35,53,48)(23,36,41,49)(26,37,27,38)(31,45,55,47)(42,61,62,58)(51,64,63,56)(52,60,54,59);(1,3,14,19,7,18,6,17)(2,8,29,33,12,32,11,13)(4,21,41,44,24,43,23,15)(5,16,26,45,28,46,27,47)(9,34,49,50,20,39,36,30)(10,31,37,57,25,55,38,40)(22,51,52,62,53,63,54,42)(35,58,59,64,48,61,60,56);(1,4,6,23,7,24,14,41)(2,9,11,36,12,20,29,49)(3,15,19,21,18,44,17,43)(5,22,27,54,28,53,26,52)(8,30,33,34,32,50,13,39)(10,35,38,60,25,48,37,59)(16,42,45,51,46,62,47,63)(31,56,57,58,55,64,40,61);(1,5)(2,10)(3,16)(4,22)(6,27)(7,28)(8,31)(9,35)(11,38)(12,25)(13,40)(14,26)(15,42)(17,47)(18,46)(19,45)(20,48)(21,51)(23,54)(24,53)(29,37)(30,56)(32,55)(33,57)(34,58)(36,60)(39,61)(41,52)(43,63)(44,62)(49,59)(50,64)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,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, 318015);