#129772

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×#3058	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

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

normal subgroups only

This group has 948 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) = 64

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)(10,17)(12,23)(13,25)(14,24)(15,26)(16,31)(18,33)(19,22)(20,35)(21,34)(27,36)(28,38)(29,39)(30,40)(32,37)

Polycyclic presentation:

2^9|a^2=f,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, 36968);