#132956

order	512
nilpotency class	4
isoclinism	isoclinic class of #132529
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 495 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40152);