#134749

order	512
nilpotency class	4
isoclinism	isoclinic class of #134748
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

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

normal subgroups only

This group has 293 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)(3,14,17,38)(4,19)(5,11,23,9)(6,25)(7,13)(8,15,29,41)(10,21)(12,26)(16,37)(18,46,45,39)(20,48,50,47)(22,49)(24,32,53,35)(27,40)(28,56)(30,58,54,42)(31,51,33,59)(34,52)(36,43)(44,60)(55,64)(57,62)(61,63);(1,3,4,15,6,18,21,42,7,17,22,41,26,45,52,58)(2,8,10,14,12,30,19,39,13,29,34,38,25,54,49,46)(5,16,20,40,24,44,51,62,23,43,50,56,53,63,59,64)(9,27,31,36,32,55,47,60,11,28,33,37,35,57,48,61);(2,9)(5,23)(8,27)(10,31)(11,13)(12,32)(14,36)(16,43)(19,47)(20,50)(24,53)(25,35)(28,29)(30,55)(33,34)(37,38)(39,60)(40,56)(44,63)(46,61)(48,49)(51,59)(54,57)(62,64)

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,f^b=fi,e^c=ei,f^c=fhi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287291, #72342994, #72343002
maximal normal subgroup of	#72287291, #72342994, #72343002

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41945);