#150017

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 243 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgh,e^b=efhi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57213);