#129310

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8300
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

621 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^c=fi,f^d=fi

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, 36506);