#133064

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	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

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

normal subgroups only

This group has 500 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,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, 40260);