#133065

order	512
nilpotency class	4
isoclinism	isoclinic class of #132541
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

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

normal subgroups only

This group has 494 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:

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

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=egi,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,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, 40261);