#391531

order	512
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2842
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14637
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

733 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298727);