#492871

order	512
nilpotency class	4
isoclinism	isoclinic class of #33404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#106
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂²×#340
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1541 conjugacy classes of subgroups; only the 864 normal subgroups are shown.

normal subgroups only

This group has 1541 conjugacy classes of subgroups — too many to draw. Only its 864 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=hi,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,f^b=fi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400067);