#476146

order	512
nilpotency class	3
isoclinism	isoclinic class of #476144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

1256 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)(5,6,10,16,12,17,11,36)(7,18,21,39,23,47,22,13)(8,24,27,48,29,14,28,42)(9,30,33,46,34,15,35,45)(19,40,51,64,52,49,25,37)(20,53,31,62,55,43,56,38)(26,41,32,44,59,57,60,61)(50,65,54,67,58,66,68,63);(5,7,12,23)(6,13,17,39)(8,25,29,51)(9,31,34,56)(10,22,11,21)(14,40,24,49)(15,43,30,53)(16,47,36,18)(19,27,52,28)(20,35,55,33)(26,58,59,50)(32,54,60,68)(37,48,64,42)(38,45,62,46)(41,67,57,63)(44,65,61,66);(5,8,12,29)(6,14,17,24)(7,19,23,52)(9,32,34,60)(10,28,11,27)(13,37,39,64)(15,44,30,61)(16,48,36,42)(18,49,47,40)(20,54,55,68)(21,25,22,51)(26,35,59,33)(31,50,56,58)(38,65,62,66)(41,46,57,45)(43,67,53,63);(2,4)(5,9)(6,15)(7,20)(8,26)(10,35)(11,33)(12,34)(13,38)(14,41)(16,46)(17,30)(18,43)(19,50)(21,56)(22,31)(23,55)(24,57)(25,54)(27,60)(28,32)(29,59)(36,45)(37,63)(39,62)(40,66)(42,61)(44,48)(47,53)(49,65)(51,68)(52,58)(64,67)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,e^2=i,g^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,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei

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, 383342);