#512805

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	#532	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

363 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:

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

Polycyclic presentation:

2^9|a^2=f,f^2=g,g^2=h,h^2=i,b^a=be,d^a=de,c^b=cei,d^b=di,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420001);