#133033

order	512
nilpotency class	4
isoclinism	isoclinic class of #132559
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	#392	C₂×C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 493 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,e^c=ei,f^c=fg,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, 40229);