#146433

order	512
nilpotency class	4
isoclinism	isoclinic class of #126701
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1735
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10721
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

853 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 853 conjugacy classes of subgroups — too many to draw. Only its 264 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53629);