#135737

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
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

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

normal subgroups only

This group has 230 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,f^b=fi,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, 42933);