#149883

order	512
nilpotency class	5
isoclinism	isoclinic class of #149800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2129
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11536
prime radical	C₂³	#413
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2961 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

204 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

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,c^b=ch,d^b=dh,e^b=efh,f^b=fi,d^c=di,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57079);