#429473

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336669);