#126543

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 170 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,21,26,22,28,39,52,23,27,40,51,41,53,69,78)(19,24,30,43,32,44,58,72,31,45,57,73,59,74,79,80)(20,33,35,61,37,49,65,50,36,62,38,67,66,25,46,48)(29,47,54,75,34,60,63,71,55,76,56,77,64,42,68,70);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,24)(20,34,36,64)(21,30)(22,32)(23,31)(25,47,49,76)(26,43)(27,45)(28,44)(29,37,55,66)(33,60,62,42)(35,63,38,68)(39,58)(40,57)(41,59)(46,54,65,56)(48,75,50,77)(51,73)(52,72)(53,74)(61,71,67,70)(69,79)(78,80);(2,7)(6,12)(8,14)(13,16)(17,20)(18,25,27,49)(19,29,31,55)(21,38)(22,37)(23,36)(24,42)(26,50,51,48)(28,33,53,62)(30,56,57,54)(32,34,59,64)(35,40)(39,46)(41,66)(43,71)(44,47)(45,60)(52,67,78,61)(58,68,79,63)(65,69)(70,73)(72,77)(74,76)(75,80)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,c^a=cgh,f^a=fi,c^b=cg,f^b=fi,d^c=di,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, 33739);