#129359

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8301
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

809 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 809 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,15,18,45,17,9,31,34,19,33,27,56,24,35,52,61)(4,20,22,54,21,39,55,42,23,37,28,57,25,10,36,38)(16,46,48,60,49,44,62,63,50,32,51,59,47,53,58,64);(1,3)(2,9)(4,16)(5,24)(6,27)(7,17)(8,19)(10,32)(11,33)(12,15)(13,34)(14,35)(18,30)(20,53)(21,49)(22,51)(23,50)(25,47)(26,31)(28,48)(29,52)(36,62)(37,44)(38,59)(39,46)(40,56)(41,45)(42,60)(43,61)(54,64)(55,58)(57,63);(1,4,7,21,8,23,5,25)(2,10,11,20,14,39,12,37)(3,16,17,49,19,50,24,47)(6,28,29,36,30,22,26,55)(9,32,33,53,35,46,15,44)(13,42,40,57,43,38,41,54)(18,51,31,58,27,48,52,62)(34,60,56,63,61,59,45,64)

Polycyclic presentation:

2^9|a^2=f,c^2=h,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,e^c=ei,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, 36555);