#130906

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 ◁ #2351 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

491 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,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, 38102);