#133244

order	512
nilpotency class	4
isoclinism	isoclinic class of #133109
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 606 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)(3,11,14,23)(4,15,17,8)(5,9)(6,10)(7,13,21,18)(12,26)(16,30,32,22)(19,24)(20,27)(25,28)(29,31)(33,34,38,41,39,44,37,43)(35,45,48,60,49,62,47,61)(36,50,51,58,52,42,53,59)(40,54,56,46,57,63,55,64);(1,3,6,14)(2,7,10,21)(4,12,17,28)(5,18,19,13)(8,20,15,29)(9,23,24,11)(16,31,32,27)(22,26,30,25)(33,35,39,49)(34,40,44,57)(36,46,52,64)(37,47,38,48)(41,55,43,56)(42,45,50,62)(51,63,53,54)(58,60,59,61);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,25)(13,27)(14,28)(18,31)(19,32)(21,29)(23,26)(24,30)(34,41)(37,38)(40,55)(42,58)(43,44)(45,60)(47,48)(50,59)(51,53)(54,63)(56,57)(61,62)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^c=fhi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40440);