#149858

order	512
nilpotency class	5
isoclinism	isoclinic class of #149792
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:

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

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,c^b=ci,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, 57054);