#130605

order	512
nilpotency class	4
isoclinism	isoclinic class of #130522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#379
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37801);