#132130

order	512
nilpotency class	4
isoclinism	isoclinic class of #132007
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39326);