#135746

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1843
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

230 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 42942);