#138181

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,6,7,5,8,4,3)(9,10)(11,21)(12,27,29,17)(13,31)(14,36)(15,19)(16,24)(18,34)(20,33)(22,44)(23,45,52,65)(25,48)(26,58)(28,61,32,63)(30,42,62,60)(35,37)(38,51,59,54)(39,49)(40,56)(41,50,43,64)(46,57)(47,55)(53,71,70,67)(66,72,68,69);(1,3,6,8,5,7,4,2)(9,11,22,49,14,26,13,24,15,25,34,57,33,55,35,56)(10,16,37,58,20,39,18,21,19,40,31,47,36,46,44,48)(12,23,50,69,30,53,32,54,29,52,64,72,62,70,28,51)(17,38,63,71,42,66,43,45,27,59,61,67,60,68,41,65);(1,4,5,6)(2,7,8,3)(9,12)(10,17)(11,23)(13,32)(14,30)(15,29)(16,38)(18,43)(19,27)(20,42)(21,45)(22,50)(24,54)(25,52)(26,53)(28,35)(31,61)(33,62)(34,64)(36,60)(37,63)(39,66)(40,59)(41,44)(46,68)(47,67)(48,65)(49,69)(51,56)(55,70)(57,72)(58,71)

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=h,d^2=ghi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45377);