#412961

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2491
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15138
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

925 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)(4,20)(5,24)(6,14)(7,12)(8,13)(9,18)(10,22)(11,26)(16,41)(17,46)(19,32)(21,51)(23,39)(25,37)(27,36)(28,38)(29,35)(30,45)(31,55)(33,44)(34,53)(40,48)(42,58)(43,49)(47,54)(50,52)(56,59)(57,61)(60,63)(62,64);(1,3,6,28,8,18,7,19)(2,9,12,38,13,15,14,32)(4,16,22,45,23,43,29,44)(5,17,25,55,27,47,26,48)(10,33,20,49,35,30,39,41)(11,31,36,46,37,40,24,54)(21,42,52,63,53,61,56,62)(34,58,50,64,51,57,59,60);(1,4)(2,10)(3,16,18,43)(5,21)(6,29)(7,22)(8,23)(9,30,15,33)(11,34)(12,39)(13,35)(14,20)(17,42,47,61)(19,45,28,44)(24,50)(25,56)(26,52)(27,53)(31,57,40,58)(32,41,38,49)(36,59)(37,51)(46,64,54,60)(48,63,55,62);(1,5,7,26,8,27,6,25)(2,11,14,24,13,37,12,36)(3,17,19,48,18,47,28,55)(4,21,22,52,23,53,29,56)(9,31,32,54,15,40,38,46)(10,34,20,50,35,51,39,59)(16,42,45,63,43,61,44,62)(30,57,41,60,33,58,49,64)

Polycyclic presentation:

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

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, 320157);