#133258

order	512
nilpotency class	4
isoclinism	isoclinic class of #133182
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2313 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 451 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=g,d^2=h,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=fh,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, 40454);