#129844

order	512
nilpotency class	4
isoclinism	isoclinic class of #129841
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×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

548 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,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, 37040);