#133769

order	512
nilpotency class	4
isoclinism	isoclinic class of #133374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 78 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)(35,45)(36,48,50,59)(37,51,53,42)(38,43)(39,44)(40,47,56,52)(41,46,58,54)(49,62)(55,63)(57,60)(61,64);(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)(33,35,39,38)(34,40,44,56)(36,46,50,54)(37,47,53,52)(41,55,58,62)(42,45,51,43)(48,61,59,57)(49,60,63,64);(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)(33,36,39,50)(34,41,44,58)(35,46,38,54)(37,49,53,63)(40,55,56,62)(42,57,51,61)(43,59,45,48)(47,60,52,64)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=eh,f^b=fg,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, 40965);