#512737

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×#2163
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

363 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ch,e^a=ei,d^b=di,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419933);