#411398

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15115
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

708 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)(4,19)(5,11)(6,28)(7,12)(8,13)(9,17)(10,23)(15,34)(16,40)(18,32)(20,47)(21,35)(22,36)(24,46)(25,53)(26,38)(27,37)(29,33)(30,49)(31,43)(39,42)(41,60)(44,57)(45,52)(48,55)(50,62)(51,59)(54,56)(58,61)(63,64);(1,3)(2,9)(4,20,23,49)(5,16)(6,29)(7,17)(8,18)(10,34,19,39)(11,31)(12,32)(13,33)(14,28)(15,24,42,22)(21,48,52,64)(25,54)(26,44)(27,43)(30,36,47,46)(35,58,45,60)(37,57)(38,56)(40,53)(41,51,61,50)(55,59,63,62);(1,4,6,22,8,23,7,24)(2,10,12,36,13,19,28,46)(3,15,17,20,18,42,29,49)(5,21,25,50,26,52,27,51)(9,30,14,39,33,47,32,34)(11,35,37,59,38,45,53,62)(16,41,43,48,44,61,54,64)(31,55,40,60,56,63,57,58);(1,5,7,27,8,26,6,25)(2,11,12,37,13,38,28,53)(3,16,17,43,18,44,29,54)(4,21,24,51,23,52,22,50)(9,31,32,57,33,56,14,40)(10,35,36,59,19,45,46,62)(15,41,20,48,42,61,49,64)(30,55,34,58,47,63,39,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318594);