#134750

order	512
nilpotency class	4
isoclinism	isoclinic class of #134747
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1812
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8608
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,14,15,37)(4,19)(5,11,23,9)(6,25)(7,13)(8,16,28,39)(10,21)(12,26)(17,36)(18,45,41,38)(20,47,49,46)(22,48)(24,32,52,35)(27,54)(29,56,53,43)(30,42)(31,50,33,59)(34,51)(40,57)(44,63)(55,62)(58,64)(60,61);(1,3,4,16,6,18,21,43,7,15,22,39,26,41,51,56)(2,8,10,14,12,29,19,38,13,28,34,37,25,53,48,45)(5,17,20,42,24,44,50,62,23,40,49,54,52,61,59,64)(9,30,31,57,32,58,46,63,11,27,33,36,35,55,47,60);(2,9)(3,15)(5,23)(8,27)(10,31)(11,13)(12,32)(14,36)(16,39)(18,41)(19,46)(20,49)(24,52)(25,35)(28,30)(29,55)(33,34)(37,57)(38,60)(43,56)(45,63)(47,48)(50,59)(53,58)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fi,e^c=ei,f^c=fhi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287362, #72342989, #72342994
maximal normal subgroup of	#72287362, #72342989, #72342994

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41946);