#150250

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
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

331 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 84 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)(9,10,11,18,16,22,15,33)(12,26,23,37,30,17,29,34)(13,31,24,21,14,32,25,20)(19,38,35,27,39,40,36,28);(1,3)(2,5)(4,8)(6,7)(10,17)(11,15)(12,27)(13,25)(14,24)(18,37)(19,21)(20,39)(22,26)(23,38)(28,30)(29,40)(31,35)(32,36)(33,34);(9,11,16,15)(10,18,22,33)(12,23,30,29)(13,24,14,25)(17,34,26,37)(19,35,39,36)(20,31,21,32)(27,40,28,38)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gh,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,f^b=fh,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, 57446);