#409328

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15016
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

803 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^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^c=gi,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, 316524);