#149857

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	#11530
prime radical	#532	C₂×C₄
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

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

normal subgroups only

This group has 242 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=h,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=efhi,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, 57053);