#434472

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15502
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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)(3,15)(4,10,24,38)(5,25)(6,13)(7,14)(8,12)(9,18)(11,28)(16,43,45,35)(17,39)(19,33)(20,34)(21,49,48,31)(22,51,54,36)(23,42,30,37)(26,32)(27,40)(29,41)(44,63,52,59)(46,58)(47,55)(50,60,64,57)(53,61,56,62);(1,3)(2,9)(4,21)(5,26)(6,19)(7,20)(8,18)(10,35)(11,39)(12,34)(13,15)(14,33)(16,23)(17,27)(22,52)(24,48)(25,55)(28,46)(29,47)(30,45)(31,37)(32,40)(36,60)(38,43)(41,58)(42,49)(44,54)(50,53)(51,57)(56,64)(59,61)(62,63);(1,4,8,30,7,24,6,23)(2,10,12,37,14,38,13,42)(3,16,18,21,20,45,19,48)(5,22,29,56,28,54,27,53)(9,31,34,43,33,49,15,35)(11,36,40,61,25,51,41,62)(17,44,26,50,47,52,46,64)(32,57,55,63,58,60,39,59);(1,5,6,27,7,28,8,29)(2,11,12,40,14,25,13,41)(3,17,18,26,20,47,19,46)(4,22,23,53,24,54,30,56)(9,32,15,39,33,58,34,55)(10,36,37,61,38,51,42,62)(16,44,21,50,45,52,48,64)(31,57,35,59,49,60,43,63)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341668);