#149563

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	C₂×#535	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

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

normal subgroups only

This group has 440 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,11,7,8,19,21,6,12,18,26,4,10,17,25)(3,9,14,23,13,20,27,30,15,24,28,32,16,22,29,31);(2,8)(4,7)(5,18)(9,22)(10,12)(11,21)(13,16)(14,28)(20,24)(23,31)(25,26)(30,32);(1,3)(2,9)(4,16)(5,14)(6,15)(7,13)(8,20)(10,22)(11,23)(12,24)(17,29)(18,28)(19,27)(21,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=e,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, 56759);