#453810

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16124
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

697 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)(3,15)(4,21,26,42)(5,27,31,11)(6,13)(7,12)(8,14)(9,18)(10,25,38,24)(16,44,47,32)(17,39,49,43)(19,35)(20,34)(22,51,46,36)(23,50)(28,33,48,59)(29,40,30,41)(37,55)(45,63)(52,62)(53,58)(54,57)(56,61)(60,64);(1,3,7,19,8,20,6,18)(2,9,13,34,14,35,12,15)(4,22,24,16,26,46,25,47)(5,28,30,17,31,48,29,49)(10,36,21,44,38,51,42,32)(11,39,41,33,27,43,40,59)(23,53,54,64,56,45,55,52)(37,58,61,62,57,63,50,60);(1,4)(2,10,14,38)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32,35,44)(11,37,27,57)(12,21,13,42)(15,36,34,51)(17,45)(18,46)(19,22)(20,47)(28,52)(29,54)(30,55)(31,56)(33,58,59,63)(39,62,43,60)(40,61,41,50)(48,64)(49,53);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,37)(12,41)(13,40)(14,27)(15,43)(16,45)(18,48)(19,28)(20,49)(21,50)(22,52)(24,55)(25,54)(26,56)(32,58)(34,39)(35,59)(36,60)(38,57)(42,61)(44,63)(46,64)(47,53)(51,62)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^c=gi,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, 361006);