#128269

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×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	#1752
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

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, 35465);