#135669

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂³	#424
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

206 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=d,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ch,d^b=di,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42865);