#132138

order	512
nilpotency class	4
isoclinism	isoclinic class of #132019
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2313 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 452 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=h,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,e^c=ei,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, 39334);