#130444

order	512
nilpotency class	4
isoclinism	isoclinic class of #130242
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62263	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

614 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37640);