#413344

order	512
nilpotency class	3
isoclinism	isoclinic class of #400512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2491
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15144
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1171 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)(3,9)(4,19,22,36)(5,24)(6,12)(7,29)(8,13)(10,23,35,28)(11,26)(14,38,20,48)(15,41)(16,32)(17,33)(18,46)(21,34,50,58)(25,37)(27,53)(30,40,56,45)(31,44)(39,55,49,64)(42,57)(43,62)(47,54,52,51)(59,63,61,60);(1,3)(2,9)(4,20,22,14)(5,15)(6,16)(7,18)(8,17)(10,30,35,56)(11,31)(12,32)(13,33)(19,38,36,48)(21,49,50,39)(23,45,28,40)(24,41)(25,42)(26,44)(27,43)(29,46)(34,55,58,64)(37,57)(47,59,52,61)(51,63,54,60)(53,62);(1,4,6,28,8,22,7,23)(2,10,12,36,13,35,29,19)(3,14,16,45,17,20,18,40)(5,21,25,54,27,50,26,51)(9,30,32,48,33,56,46,38)(11,34,24,52,37,58,53,47)(15,39,42,63,43,49,44,60)(31,55,41,61,57,64,62,59);(1,5,7,26,8,27,6,25)(2,11,12,24,13,37,29,53)(3,15,18,44,17,43,16,42)(4,21,23,51,22,50,28,54)(9,31,32,41,33,57,46,62)(10,34,36,52,35,58,19,47)(14,39,40,60,20,49,45,63)(30,55,48,61,56,64,38,59)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=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, 320540);