#128284

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

321 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

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

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, 35480);