#452104

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16029
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 359300);