#149959

order	512
nilpotency class	5
isoclinism	isoclinic class of #149958
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11560
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 ◁ #2930 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

283 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57155);