#453860

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	#16161
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

715 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:

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

Polycyclic presentation:

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

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, 361056);