#129888

order	512
nilpotency class	4
isoclinism	isoclinic class of #129781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

751 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^c=fi,f^d=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, 37084);