#93507

order	512
nilpotency class	4
isoclinism	isoclinic class of #7307
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	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×C₄×M₁₆	C₂²
socle	C₂²	#1280
prime radical	C₂⁴	#97
LCS	G ▷ C₂²×C₄ ▷ 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)(9,10,13,18,15,17,28,34,14,19,29,36,12,24,26,35)(11,20,22,37,21,16,30,31,23,33,27,39,25,32,38,40);(1,2,3,6,4,5,7,8)(9,11)(10,16)(12,25)(13,27)(14,23)(15,21)(17,33)(18,31)(19,32)(20,24)(22,29)(26,30)(28,38)(34,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=h,e^2=gi,h^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fi,c^b=cg,d^b=dfi,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, 703);