#450835

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15979
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

934 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)(3,14,18,9)(4,19)(5,11)(6,27)(7,13)(8,12)(10,22)(15,42,45,34)(16,31,47,38)(17,32,28,33)(20,30,44,41)(21,48)(23,37)(24,52)(25,40)(26,39)(29,36)(35,50)(43,58,64,59)(46,57,53,56)(49,62,63,55)(51,61)(54,60);(1,3,6,17,8,18,7,28)(2,9,27,33,12,14,13,32)(4,20,22,15,23,44,29,45)(5,16,24,46,26,47,25,53)(10,34,37,30,36,42,19,41)(11,38,52,56,39,31,40,57)(21,49,50,43,51,63,54,64)(35,59,61,62,60,58,48,55);(1,4)(2,10,12,36)(3,15)(5,21)(6,22)(7,29)(8,23)(9,30,14,41)(11,35,39,60)(13,19,27,37)(16,43)(17,44)(18,45)(20,28)(24,50)(25,54)(26,51)(31,55,38,62)(32,34,33,42)(40,48,52,61)(46,63)(47,64)(49,53)(56,58,57,59);(1,5,6,24,8,26,7,25)(2,11,27,52,12,39,13,40)(3,16,17,46,18,47,28,53)(4,21,22,50,23,51,29,54)(9,31,33,57,14,38,32,56)(10,35,37,61,36,60,19,48)(15,43,44,63,45,64,20,49)(30,55,42,59,41,62,34,58)

Polycyclic presentation:

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

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, 358031);