#471163

order	512
nilpotency class	3
isoclinism	isoclinic class of #459299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2752
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16768
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

620 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378359);