#150644

order	512
nilpotency class	5
isoclinism	isoclinic class of #150564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

365 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,20,25,22,27,35,43)(19,28,31,41,30,39,45,48)(21,26,36,42,29,24,34,40)(23,33,38,44,37,46,47,32);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,29,44,22,30,21,32)(18,23,26,39,27,37,24,28)(20,33,36,45,35,46,34,31)(25,41,40,47,43,48,42,38);(2,7)(6,14)(8,15)(9,16)(19,30)(23,37)(28,39)(31,45)(32,44)(33,46)(38,47)(41,48)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,e^d=ehi,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, 57840);