#151000

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂³	#357
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

192 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,e^2=fghi,g^2=hi,h^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58196);