#149947

order	512
nilpotency class	5
isoclinism	isoclinic class of #149945
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11557
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	#525	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 395 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57143);