#445100

order	512
nilpotency class	3
isoclinism	isoclinic class of #423438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15802
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

680 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)(5,25)(6,29)(7,13)(8,12)(10,22)(11,28)(15,35,42,57)(16,38,44,32)(17,33,30,34)(20,40,48,31)(21,49)(23,37)(24,46)(26,39)(27,54)(36,51)(41,60,63,61)(43,58,55,59)(45,52)(47,62,64,56)(50,53);(1,3,6,17,8,18,7,30)(2,9,29,34,12,14,13,33)(4,20,22,42,24,48,23,15)(5,16,26,43,28,44,27,55)(10,35,46,40,37,57,19,31)(11,38,54,58,25,32,39,59)(21,47,50,63,52,64,51,41)(36,61,49,56,53,60,45,62);(1,4,7,23,8,24,6,22)(2,10,29,46,12,37,13,19)(3,15,17,20,18,42,30,48)(5,21,27,51,28,52,26,50)(9,31,33,57,14,40,34,35)(11,36,54,49,25,53,39,45)(16,41,43,47,44,63,55,64)(32,56,58,61,38,62,59,60);(1,5)(2,11,12,25)(3,16)(4,21)(6,26)(7,27)(8,28)(9,32,14,38)(10,36,37,53)(13,39,29,54)(15,41)(17,43)(18,44)(19,45,46,49)(20,47)(22,50)(23,51)(24,52)(30,55)(31,56,40,62)(33,58,34,59)(35,60,57,61)(42,63)(48,64)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=eg,d^2=e,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,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, 352296);