#479325

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17199
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

642 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)(3,15,20,34)(4,21)(5,27)(6,12)(7,13)(8,14)(9,18,35,19)(10,25)(11,30)(16,32,48,53)(17,33,52,40)(22,44,47,36)(23,37)(24,39)(26,38)(28,41)(29,43)(31,42)(45,51,57,50)(46,59,63,60)(49,64,54,58)(55,61)(56,62);(1,3)(2,9,14,35)(4,22)(5,17)(6,18)(7,19)(8,20)(10,36,39,44)(11,40,43,33)(12,15,13,34)(16,24)(21,53,38,32)(23,54)(25,48)(26,47)(27,57,42,45)(28,46)(29,50)(30,51)(31,52)(37,60,62,59)(41,58,61,64)(49,56)(55,63);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,41)(12,38)(13,21)(14,39)(15,44)(17,49)(18,22)(19,47)(20,48)(23,29)(27,37)(30,56)(31,55)(33,59)(34,36)(35,53)(40,60)(42,62)(43,61)(45,58)(46,50)(51,63)(52,54)(57,64);(1,5,7,30,8,31,6,29)(2,11,12,27,14,43,13,42)(3,17,19,51,20,52,18,50)(4,23,25,55,26,56,24,28)(9,33,15,45,35,40,34,57)(10,37,21,41,39,62,38,61)(16,46,22,54,48,63,47,49)(32,58,36,59,53,64,44,60)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,d^2=f,e^2=i,f^2=i,g^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,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386521);