#141810

order	512
nilpotency class	4
isoclinism	isoclinic class of #141698
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 271 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,22,29,23,27,24,46)(19,25,33,49,31,50,44,51)(20,34,37,67,38,66,45,68)(21,39,41,57,42,75,43,28)(26,52,55,36,54,72,56,73)(30,59,35,65,60,80,62,79)(32,63,64,48,40,74,69,58)(47,71,77,61,78,70,53,76);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,24,33,23,31,22,44)(18,25,29,51,27,50,46,49)(20,35,45,60,38,62,37,30)(21,40,43,69,42,32,41,64)(26,53,55,78,54,77,56,47)(28,58,39,74,57,48,75,63)(34,65,67,59,66,79,68,80)(36,71,52,61,73,70,72,76);(3,11)(7,15)(10,16)(12,13)(17,20,21,36,24,37,43,72,23,38,42,73,22,45,41,52)(18,26,28,34,29,56,39,68,27,54,57,66,46,55,75,67)(19,30,32,61,33,62,64,71,31,60,40,76,44,35,69,70)(25,47,48,59,51,77,63,65,50,78,58,80,49,53,74,79)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=e,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cgh,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210683, #72210684
maximal normal subgroup of	#72210683, #72210684
maximal quotient of	#72210531, #72210560, #72210570

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49006);