#129774

order	512
nilpotency class	4
isoclinism	isoclinic class of #129769
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	C₂×#3061	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 108 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)(9,10,13,20,15,19,14,21)(11,22,24,35,26,17,25,34)(12,27,29,18,30,16,28,32)(23,31,38,33,40,36,39,37);(1,3)(2,5)(4,8)(6,7)(10,16)(12,28)(13,14)(17,31)(18,20)(19,27)(21,32)(22,36)(23,38)(24,25)(29,30)(33,35)(34,37)(39,40);(9,11,15,26)(10,17,19,22)(12,23,30,40)(13,25,14,24)(16,31,27,36)(18,33,32,37)(20,35,21,34)(28,38,29,39)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36970);