#494149

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1843
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265417, #72265441, #72265447
maximal normal subgroup of	#72265417, #72265441, #72265447

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401345);