#489378

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62432	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

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, 396574);