#512157

order	512
nilpotency class	4
isoclinism	isoclinic class of #511785
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1047 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,d^b=di,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419353);