#134612

order	512
nilpotency class	4
isoclinism	isoclinic class of #124661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1545
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8588
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265701, #72265706, #72265723
maximal normal subgroup of	#72265701, #72265706, #72265723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41808);