#442605

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15722
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

735 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)(4,19)(5,23,26,52)(6,28)(7,12)(8,13)(9,17)(10,22)(11,27,39,25)(15,41)(16,46,24,53)(18,33)(20,35,49,59)(21,37)(29,34)(30,36)(31,44)(32,55,38,47)(40,43)(42,57,54,60)(45,58)(48,50,51,56)(61,63,64,62);(1,3,6,29,8,17,7,18)(2,9,12,34,13,14,28,33)(4,15,30,43,21,45,22,44)(5,24,27,55,26,16,25,47)(10,31,37,58,36,40,19,41)(11,38,52,53,39,32,23,46)(20,42,50,63,49,54,56,62)(35,57,48,61,59,60,51,64);(1,4)(2,10)(3,15)(5,20)(6,30)(7,22)(8,21)(9,31)(11,35)(12,37)(13,36)(14,40)(16,42)(17,45)(18,44)(19,28)(23,51)(24,54)(25,56)(26,49)(27,50)(29,43)(32,57)(33,41)(34,58)(38,60)(39,59)(46,61)(47,63)(48,52)(53,64)(55,62);(1,5,6,25,8,26,7,27)(2,11,12,23,13,39,28,52)(3,16,29,55,17,24,18,47)(4,20,30,56,21,49,22,50)(9,32,34,53,14,38,33,46)(10,35,37,51,36,59,19,48)(15,42,43,62,45,54,44,63)(31,57,58,64,40,60,41,61)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^d=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, 349801);