#133768

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×#540	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#540 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 628 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^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, 40964);