#133003

order	512
nilpotency class	4
isoclinism	isoclinic class of #132501
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 495 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40199);