#129106

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^b=ch,d^b=dg,e^b=eg,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, 36302);