#471283

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16786
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

658 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)(3,8,16,27)(4,18,21,45)(5,10,23,32)(6,12)(7,11)(9,20,30,35)(13,28)(14,37)(15,25)(17,26,34,54)(19,46)(22,33,36,55)(24,49)(29,47)(31,56)(38,61,41,59)(39,60)(40,51)(42,53)(43,52)(44,48)(50,63,57,62)(58,64);(1,3,13,34,7,16,6,17)(2,8,12,26,11,27,28,54)(4,14,20,40,21,39,35,49)(5,15,36,58,23,42,22,43)(9,24,45,60,30,51,18,37)(10,25,33,52,32,53,55,64)(19,38,47,62,48,41,56,63)(29,50,46,61,31,57,44,59);(1,4,6,20,7,21,13,35)(2,9,28,45,11,30,12,18)(3,14,34,49,16,39,17,40)(5,19,22,47,23,48,36,56)(8,24,26,37,27,51,54,60)(10,31,55,44,32,29,33,46)(15,41,58,62,42,38,43,63)(25,50,52,59,53,57,64,61);(1,5,6,22,7,23,13,36)(2,10,12,33,11,32,28,55)(3,15,17,43,16,42,34,58)(4,19,20,47,21,48,35,56)(8,25,26,52,27,53,54,64)(9,29,18,44,30,31,45,46)(14,38,40,62,39,41,49,63)(24,50,37,59,51,57,60,61)

Polycyclic presentation:

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

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, 378479);