#93505

order	512
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator, prime radical	C₂⁴	C₄×C₈
Frattini, cosocle	C₂×C₄×M₁₆	C₂²
socle	C₂²	#1280
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

255 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=h,e^2=gi,h^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fi,c^b=cg,d^b=df,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 701);