#129107

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

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^a=ci,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, 36303);