#129728

order	512
nilpotency class	4
isoclinism	isoclinic class of #124660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1510
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8328
prime radical	#60453	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

863 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 863 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36924);