#355093

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13775
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1861 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)(3,15,19,9)(4,20,23,39)(5,24,26,11)(6,13)(7,14)(8,12)(10,22,35,29)(16,41)(17,31)(18,38,28,32)(21,50)(25,37,27,36)(30,43)(33,44)(34,51)(40,47)(42,61,63,60)(45,48)(46,57)(49,52)(53,55)(54,59)(56,62,58,64);(1,3,7,18,8,19,6,28)(2,9,14,32,12,15,13,38)(4,16,22,44,23,45,29,43)(5,17,27,46,26,47,25,54)(10,33,20,48,35,30,39,41)(11,31,37,57,24,40,36,59)(21,42,51,64,52,63,55,62)(34,58,49,61,53,56,50,60);(1,4)(2,10)(3,16)(5,21)(6,29)(7,22)(8,23)(9,30)(11,34)(12,35)(13,39)(14,20)(15,33)(17,42)(18,44)(19,45)(24,53)(25,55)(26,52)(27,51)(28,43)(31,56)(32,41)(36,50)(37,49)(38,48)(40,58)(46,64)(47,63)(54,62)(57,60)(59,61);(1,5)(2,11)(3,17)(4,21)(6,25)(7,27)(8,26)(9,31)(10,34)(12,24)(13,36)(14,37)(15,40)(16,42)(18,46)(19,47)(20,49)(22,51)(23,52)(28,54)(29,55)(30,56)(32,57)(33,58)(35,53)(38,59)(39,50)(41,60)(43,62)(44,64)(45,63)(48,61)

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,f^b=fi,g^b=gi,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262289);