#129807

order	512
nilpotency class	4
isoclinism	isoclinic class of #129741
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 800 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,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, 37003);