#433020

order	512
nilpotency class	3
isoclinism	isoclinic class of #433019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340216);