#404718

order	512
nilpotency class	3
isoclinism	isoclinic class of #397903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2856
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14900
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1157 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1157 conjugacy classes of subgroups — too many to draw. Only its 204 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)(3,15)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,28)(16,31)(17,46)(19,34)(20,33)(22,35)(23,52)(24,38)(26,37)(29,39)(30,40)(32,49)(36,55)(41,44)(42,47)(43,61)(45,57)(48,58)(50,54)(51,56)(53,60)(59,63)(62,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,16,26,44,25,45)(5,17,28,47,29,49,30,48)(10,35,37,57,38,41,21,31)(11,32,27,42,40,46,39,58)(23,51,53,43,55,62,54,63)(36,59,60,64,52,61,50,56);(1,4,6,25,8,26,7,24)(2,10,12,21,14,38,13,37)(3,16,19,44,20,45,18,22)(5,23,28,54,29,55,30,53)(9,31,33,35,34,57,15,41)(11,36,27,50,40,52,39,60)(17,43,48,62,49,63,47,51)(32,56,58,59,46,64,42,61);(1,5,6,28,8,29,7,30)(2,11,13,39,14,40,12,27)(3,17,18,47,20,49,19,48)(4,23,25,54,26,55,24,53)(9,32,33,58,34,46,15,42)(10,36,37,60,38,52,21,50)(16,43,22,51,45,63,44,62)(31,56,35,59,57,64,41,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311914);