#150789

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33733	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 236 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=fhi,d^2=gh,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=cfi,d^b=dg,e^b=efgh,f^b=fh,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, 57985);