#512152

order	512
nilpotency class	4
isoclinism	isoclinic class of #511779
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	#3484	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1126 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1126 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)(19,29)(20,27)(21,28)(22,30)(23,32)(24,37)(25,31)(26,34)(33,40)(35,39)(36,38);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,28)(17,27)(18,29)(22,35)(23,33)(24,36)(25,34)(26,31)(30,39)(32,40)(37,38);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,26)(17,30)(18,31)(20,33)(21,34)(27,38)(28,29)(32,37)(35,36)(39,40);(9,12,14,23)(10,17,18,32)(11,20,19,33)(13,24,25,22)(15,27,29,40)(16,30,31,37)(21,36,34,35)(26,38,28,39)

Polycyclic presentation:

2^9|c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,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, 419348);