#150975

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
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	#2215	C₄
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

205 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58171);