#472529

order	512
nilpotency class	3
isoclinism	isoclinic class of #457680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2665
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16823
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

843 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)(3,13,17,8)(4,9)(5,10,24,34)(6,12)(7,11)(14,26)(15,27,43,36)(16,37)(18,38,39,29)(19,30,48,58)(20,31,23,50)(21,33)(22,32)(25,35,41,54)(28,46)(40,53)(42,55)(44,60)(45,61)(47,57)(49,59,52,64)(51,63)(56,62);(1,3,14,39,7,17,6,18)(2,8,26,38,11,13,12,29)(4,19,22,43,21,48,40,15)(5,16,41,62,24,46,25,45)(9,30,32,36,33,58,53,27)(10,28,54,61,34,37,35,56)(20,47,49,44,23,51,52,42)(31,57,59,60,50,63,64,55);(1,4)(2,9)(3,15,17,43)(5,23,24,20)(6,22)(7,21)(8,27,13,36)(10,31,34,50)(11,33)(12,32)(14,40)(16,44)(18,19,39,48)(25,52,41,49)(26,53)(28,55)(29,30,38,58)(35,59,54,64)(37,60)(42,46)(45,51)(47,62)(56,57)(61,63);(1,5)(2,10)(3,16)(4,20)(6,25)(7,24)(8,28)(9,31)(11,34)(12,35)(13,37)(14,41)(15,42)(17,46)(18,45)(19,47)(21,23)(22,49)(26,54)(27,55)(29,56)(30,57)(32,59)(33,50)(36,60)(38,61)(39,62)(40,52)(43,44)(48,51)(53,64)(58,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379725);