#129727

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

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

normal subgroups only

This group has 871 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)(3,14,17,42)(4,20,22,10)(5,25,28,57)(6,13)(7,31)(8,12)(9,16)(11,29,39,30)(15,41)(18,37,19,50)(21,53)(23,52,24,36)(26,40)(27,59)(32,43,51,47)(33,49)(34,48)(35,55)(38,56)(44,60)(45,64)(46,62)(54,61,58,63);(1,3)(2,9)(4,15)(5,26)(6,19)(7,18)(8,17)(10,32)(11,37)(12,34)(13,40)(14,25)(16,30)(20,51)(21,54)(22,44)(23,46)(24,45)(27,58)(28,49)(29,48)(31,33)(35,62)(36,63)(38,64)(39,50)(41,53)(42,57)(43,56)(47,55)(52,61)(59,60);(1,4)(2,10)(3,15)(5,27)(6,23)(7,24)(8,22)(9,32)(11,38)(12,20)(13,36)(14,41)(16,47)(17,44)(18,45)(19,46)(21,28)(25,53)(26,58)(29,56)(30,55)(31,52)(33,61)(34,51)(35,39)(37,64)(40,63)(42,60)(43,48)(49,54)(50,62)(57,59)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,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, 36923);