#489198

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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	#61412	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396394);