#129825

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 769 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)(11,20)(12,25,27,17)(13,18)(14,19)(15,22,32,21)(16,26,33,28)(23,37)(24,34)(29,35)(30,40)(31,39,36,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15,19,32)(12,21)(13,23)(14,24)(16,30,33,37)(17,34,25,20)(18,31,35,36)(22,27)(26,38)(28,39)(29,40);(9,12)(10,16)(11,21)(13,28)(14,27)(15,30)(17,18)(19,33)(20,36)(22,24)(23,39)(25,35)(26,29)(31,34)(32,37)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fghi,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, 37021);