#432368

order	512
nilpotency class	3
isoclinism	isoclinic class of #418473
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15456
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

869 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)(3,14,18,33)(4,10)(5,23)(6,12)(7,28)(8,13)(9,17,32,27)(11,25)(15,42,45,41)(16,46,47,31)(19,49)(20,36)(21,54)(22,37)(24,56,48,38)(26,39)(29,40)(30,53,34,44)(35,51)(43,57,63,58)(50,62,64,59)(52,60)(55,61);(1,3,6,27,8,18,7,17)(2,9,12,33,13,32,28,14)(4,15,20,53,22,45,21,44)(5,24,25,16,26,48,29,47)(10,34,36,41,37,30,54,42)(11,38,39,31,40,56,23,46)(19,50,51,43,52,64,55,63)(35,59,60,58,61,62,49,57);(1,4,6,20,8,22,7,21)(2,10,12,36,13,37,28,54)(3,15,27,53,18,45,17,44)(5,19,25,51,26,52,29,55)(9,30,33,42,32,34,14,41)(11,35,39,60,40,61,23,49)(16,43,48,64,47,63,24,50)(31,57,56,59,46,58,38,62);(1,5)(2,11)(3,16,18,47)(4,19)(6,25)(7,29)(8,26)(9,31,32,46)(10,35)(12,39)(13,40)(14,38,33,56)(15,43,45,63)(17,24,27,48)(20,51)(21,55)(22,52)(23,28)(30,57,34,58)(36,60)(37,61)(41,62,42,59)(44,50,53,64)(49,54)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=ef,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 339564);