#149905

order	512
nilpotency class	5
isoclinism	isoclinic class of #149903
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	#382	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#382 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 315 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=dh,e^b=efhi,f^b=fi,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, 57101);