#149943

order	512
nilpotency class	5
isoclinism	isoclinic class of #149893
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2122
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	#11557
prime radical	#24285	C₂
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 406 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cg,e^b=efi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57139);