#150875

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
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

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

normal subgroups only

This group has 286 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) = 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)(9,10,12,19,15,23,30,22)(11,24,27,38,29,16,34,35)(13,32,31,20,14,33,26,21)(17,39,36,28,18,40,37,25);(1,3)(2,5)(4,8)(6,7)(10,16)(11,25)(12,30)(13,31)(14,26)(17,20)(18,21)(19,38)(22,35)(23,24)(27,40)(28,29)(32,36)(33,37)(34,39);(3,8)(5,7)(9,11,26,39,12,27,13,28,15,29,31,40,30,34,14,25)(10,17,33,35,22,37,20,16,23,18,32,38,19,36,21,24)

Polycyclic presentation:

2^9|a^2=e,c^2=gh,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=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, 58071);