#150773

order	512
nilpotency class	5
isoclinism	isoclinic class of #150443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	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 ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 54 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)(17,18,20,25,22,27,33,44)(19,28,31,46,30,39,47,43)(21,26,37,45,36,24,34,42)(23,38,40,29,41,32,48,35);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,22,30)(18,23)(20,32,33,38)(21,35,36,29)(24,28)(25,43)(26,39)(27,41)(31,34,47,37)(40,42)(44,46)(45,48);(2,7)(6,14)(8,15)(9,16)(19,29)(20,33)(21,36)(23,39)(24,26)(25,44)(28,41)(30,35)(31,38)(32,47)(40,46)(43,48)

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=gi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfh,d^b=dg,e^b=efg,f^b=fi,d^c=dhi,e^c=ei,f^c=fi,e^d=ehi,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, 57969);