#502777

order	512
nilpotency class	4
isoclinism	isoclinic class of #501164
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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	#3496	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=ei,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409973);