#493489

order	512
nilpotency class	4
isoclinism	isoclinic class of #493483
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#62887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #3486 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2142 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400685);