#409540

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15026
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

907 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)(3,15,19,33)(4,20)(5,26,29,11)(6,12)(7,13)(8,14)(9,18,34,30)(10,24)(16,31,43,57)(17,41)(21,35,49,56)(22,47,52,60)(23,38)(25,37)(27,40,28,39)(32,44)(36,51,54,50)(42,61)(45,53)(46,58)(48,64)(55,63)(59,62);(1,3)(2,9)(4,21,25,49)(5,17)(6,30)(7,18)(8,19)(10,35,38,56)(11,32)(12,33)(13,15)(14,34)(16,24,43,23)(20,31,37,57)(22,48,52,62)(26,53)(27,45)(28,44)(29,46)(36,59,54,64)(39,41)(40,58)(42,50,63,51)(47,55,60,61);(1,4)(2,10,14,38)(3,16,19,43)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36,26,54)(12,20,13,37)(15,35)(17,42,46,63)(18,21,30,49)(27,50)(28,51)(29,52)(32,55)(33,56)(34,57)(39,60,40,47)(41,59)(44,62,45,48)(53,61)(58,64);(1,5)(2,11)(3,17)(4,22)(6,28)(7,27)(8,29)(9,32)(10,36)(12,40)(13,39)(14,26)(15,41)(16,42)(18,45)(19,46)(20,47)(21,48)(23,51)(24,50)(25,52)(30,44)(31,55)(33,58)(34,53)(35,59)(37,60)(38,54)(43,63)(49,62)(56,64)(57,61)

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,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, 316736);