#150873

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

289 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=ghi,d^2=gh,e^2=i,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=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58069);