#411705

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,24,37)(8,25,29,44)(9,30,33,42)(10,17,11,16)(13,23,38,22)(14,34,41,28)(15,31,43,32)(20,47,50,35)(21,51,54,36)(26,39,49,45)(27,57,59,40)(46,53,62,52)(48,63,68,65)(55,60,64,58)(56,61,67,66);(2,4)(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,39)(15,36)(16,37)(17,19)(18,38)(20,28)(25,35)(27,56)(29,49)(30,46)(31,53)(32,52)(33,54)(34,50)(40,63)(41,45)(42,62)(43,51)(44,47)(48,58)(55,61)(57,65)(59,67)(60,68)(64,66);(5,8)(6,14)(7,20)(9,27)(10,34)(11,28)(12,29)(13,35)(15,40)(16,44)(17,25)(18,41)(19,45)(21,48)(22,49)(23,26)(24,50)(30,55)(31,60)(32,58)(33,59)(36,61)(37,39)(38,47)(42,64)(43,57)(46,65)(51,66)(52,67)(53,56)(54,68)(62,63);(5,9,10,31,12,33,11,32)(6,15,16,42,18,43,17,30)(7,21,23,53,24,54,22,52)(8,27,34,60,29,59,28,58)(13,36,37,62,38,51,19,46)(14,40,44,64,41,57,25,55)(20,48,26,56,50,68,49,67)(35,61,39,63,47,66,45,65)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,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, 318901);