#149723

order	512
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11481
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,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^d=ehi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56919);