#411416

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15116
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

607 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=e,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,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, 318612);