#498757

order	512
nilpotency class	3
isoclinism	isoclinic class of #31077
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23352
prime radical	#532	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,c^b=cefh,d^b=de,f^b=fi,g^b=gi,d^c=di

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, 405953);