#138947

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

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

normal subgroups only

This group has 518 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=gh,e^2=gh,f^2=gh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46143);