#149708

order	512
nilpotency class	5
isoclinism	isoclinic class of #13595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11477
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

261 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

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=ci,d^b=dh,e^b=efh,f^b=fi,e^d=ehi,f^e=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, 56904);