#135179

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	C₂×#323
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^c=ei,f^c=fgi,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42375);