#133820

order	512
nilpotency class	4
isoclinism	isoclinic class of #133434
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2313 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 473 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,e^c=ei,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, 41016);