#150637

order	512
nilpotency class	5
isoclinism	isoclinic class of #150567
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 375 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)(19,28,31,23)(20,24)(21,26,29,25)(22,27)(30,41,44,35)(32,36,42,37)(33,39,43,38)(34,40)(45,46,48,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,29,42,22,32,21,31)(18,23,26,36,27,37,25,28)(20,30,43,48,34,45,33,44)(24,35,39,46,40,47,38,41);(2,7)(6,14)(8,15)(9,16)(17,20,22,34)(18,24,27,40)(19,30,32,45)(21,33,29,43)(23,35,37,47)(25,38,26,39)(28,41,36,46)(31,44,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57833);