#128213

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,3,6,4,5,7,8)(9,10,14,21,13,23,38,22,17,24,40,54,16,20,15,41)(11,25,28,56,27,18,42,44,31,46,63,45,30,47,29,64)(12,32,34,66,33,52,35,53,37,49,39,51,36,19,48,50)(26,43,58,68,57,65,59,72,61,70,62,71,60,55,67,69);(9,11,15,29,16,30,40,63,17,31,38,42,13,27,14,28)(10,18,22,45,20,25,21,44,24,47,41,56,23,46,54,64)(12,26,35,59,36,60,34,58,37,61,48,67,33,57,39,62)(19,43,51,69,49,55,53,71,52,70,66,72,32,65,50,68);(2,5)(6,8)(9,12,13,33,17,37,16,36)(10,19,20,49,24,52,23,32)(11,26,27,57,31,61,30,60)(14,39,38,48,40,34,15,35)(18,43,25,55,47,70,46,65)(21,53,41,66,54,50,22,51)(28,62,42,67,63,58,29,59)(44,71,56,72,64,68,45,69)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35409);