#455831

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16270
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1069 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)(5,27)(6,13)(7,12)(8,14)(9,18)(10,24)(11,30)(16,45)(17,33)(19,34)(20,41)(22,43)(23,36)(25,37)(26,42)(28,53)(29,40)(31,39)(32,48)(35,47)(38,55)(44,50)(46,58)(49,57)(51,59)(52,60)(54,62)(56,61)(63,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,41)(4,22,24,48,26,47,25,16)(5,17,30,51,31,52,29,50)(10,35,21,32,37,43,42,45)(11,33,27,44,40,60,39,59)(23,54,55,64,56,49,28,46)(36,58,53,57,61,63,38,62);(1,4,6,24,8,26,7,25)(2,10,12,21,14,37,13,42)(3,16,18,22,20,48,19,47)(5,28,29,56,31,55,30,23)(9,32,15,43,34,45,41,35)(11,38,39,61,40,53,27,36)(17,49,50,64,52,54,51,46)(33,58,59,62,60,63,44,57);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,36)(12,27)(13,39)(14,40)(15,44)(16,46)(18,51)(19,50)(20,52)(21,53)(22,54)(24,55)(25,28)(26,56)(32,57)(34,60)(35,58)(37,61)(38,42)(41,59)(43,63)(45,62)(47,49)(48,64)

Polycyclic presentation:

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

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, 363027);