#496097

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21683
prime radical	#23794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 678 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224961, C₂×#496097, #72320755
maximal normal subgroup of	#72224961, C₂×#496097, #72320755
maximal quotient of	C₂×#496097
automorphism group of	#7511

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403293);