#471534

order	512
nilpotency class	3
isoclinism	isoclinic class of #454389
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2815
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16799
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 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)(3,14)(4,10)(5,25)(6,30)(7,12)(8,13)(9,17)(11,27)(15,34)(16,47)(18,43)(19,33)(20,52)(21,54)(22,37)(23,36)(24,38)(26,58)(28,57)(29,40)(31,45)(32,51)(35,56)(39,55)(41,46)(42,49)(44,59)(48,62)(50,63)(53,60)(61,64);(1,3)(2,9,13,33)(4,20,24,45)(5,16)(6,17)(7,19)(8,18)(10,34)(11,32,40,47)(12,14,30,43)(15,23,46,22)(21,53,55,48)(25,42,57,63)(26,44,56,64)(27,49)(28,51)(29,50)(31,37)(35,60)(36,52)(38,41)(39,61)(54,59)(58,62);(1,4)(2,10)(3,15)(5,26)(6,23)(7,22)(8,24)(9,31)(11,39)(12,37)(13,38)(14,41)(16,48)(17,20)(18,46)(19,45)(21,27)(25,35)(28,56)(29,55)(30,36)(32,60)(33,52)(34,43)(40,54)(42,59)(44,49)(47,62)(50,64)(51,53)(57,58)(61,63);(1,5,6,27,8,28,7,29)(2,11,12,25,13,40,30,57)(3,16,17,49,18,51,19,50)(4,21,22,26,24,55,23,56)(9,32,14,42,33,47,43,63)(10,35,36,39,38,58,37,54)(15,44,45,48,46,64,20,53)(31,59,34,60,52,61,41,62)

Polycyclic presentation:

2^9|b^2=gh,d^2=e,e^2=i,g^2=i,h^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,g^c=gi,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, 378730);