#138948

order	512
nilpotency class	4
isoclinism	isoclinic class of #138944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9259
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#33522	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281073
maximal normal subgroup of	#72281073

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46144);