#149867

order	512
nilpotency class	5
isoclinism	isoclinic class of #149791
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11532
prime radical	C₂³	#413
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2960 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=dh,e^b=efh,f^b=fi,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57063);