#471462

order	512
nilpotency class	3
isoclinism	isoclinic class of #471461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318258, #72318890, #72319003
maximal normal subgroup of	#72318258, #72318890, #72319003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378658);