#130243

order	512
nilpotency class	4
isoclinism	isoclinic class of #130242
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#62258	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 612 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,f^2=h,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, 37439);