#135682

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

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

196 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:

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

Polycyclic presentation:

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

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, 42878);