#128807

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1514
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8216
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

382 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,d^c=di,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36003);