#127614

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1505
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8022
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,g^b=gi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34810);