#405285

order	512
nilpotency class	3
isoclinism	isoclinic class of #405284
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14911
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2159 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312481);