#129116

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1568
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,d^b=dg,e^b=egi,g^b=gi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36312);