#149564

order	512
nilpotency class	5
isoclinism	isoclinic class of #149559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11426
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	#407	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2879 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56760);