#133735

order	512
nilpotency class	4
isoclinism	isoclinic class of #133293
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
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 ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

609 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgi,d^c=dh,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, 40931);