#136832

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8965
prime radical	#3018	C₄
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

491 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=egi,g^a=gi,c^b=ci,e^b=egi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343161, #72343184, #72343185
maximal normal subgroup of	#72343161, #72343184, #72343185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44028);